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NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE PDF Download

SHORT ANSWER TYPE QUESTIONS

Q.1. Find the unit vector in the direction of sum of vectors
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
Given that
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ Unit vector in the direction ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required unit vector isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.2. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEfind the unit vector in the direction of
(i) NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(ii)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(i) NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ Unit vector in the direction ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required unit vector isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(ii)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE-NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ Unit vector in the direction ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required unit vector isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.3. Find a unit vector in the direction ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEwhere P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively.
Ans.
Given coordinates are P(5, 0, 8) and Q(3, 3, 2)
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ Unit vector in the direction ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required unit vector isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.4. If NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.
Ans.
Given that
BC = 1.5 BA
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required vector isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.5. Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.
Ans.
Let the given points are A( k , - 10 , 3), B(1,- 1, 3) and C(3, 5, 3)
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

If A, B and C are collinear, then

NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Squaring both sides, we have
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
= 9 + k2- 6k + 225
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Dividing by 2, we get
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE(Dividing by 2)
Squaring both sides, we get
⇒ 10(k2 – 2k + 82) = 784 + k2 – 56k
⇒ 10k2 – 20k + 820 = 784 + k2 – 56k
⇒ 10k2 – k2 – 20k + 56k + 820 – 784 = 0
⇒ 9k2 + 36k + 36 = 0
⇒ k2 + 4k + 4 = 0
⇒ (k + 2)2 = 0
⇒ k + 2 = 0
⇒ k = – 2
Hence, the required value is k = – 2

Q.6. A vectorNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis inclined at equal angles to the three axes. If the magnitude of NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE is 2 √3 units, findNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
Since, the vectorNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEmakes equal angles with the axes, their direction cosines should be same
∴ l = m = n
We know that
l2 + m2 + n2 = 1
⇒ l2 + l2 + l2 = 1
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
We know thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required value ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.7. A vector NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEhas magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE , given that NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEmakes an acute angle with x-axis.
Ans.
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEbe three vectors such thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEandNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
If l, m and n are the direction cosines of vector NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE, then
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
We know that l2 + m2 + n= 1
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ k = ± 2 and l =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required direction cosines areNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEand the components of
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.8. Find a vector of magnitude 6, which is perpendicular to both the vectors NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEandNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEandNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
We know that unit vector perpendicular toNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Now the vector of magnitude 6 =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required vector isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.9. Find the angle between the vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE.
Ans.
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE and let θ be the angle betweenNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required value of θ isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.10. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE show that NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE  Interpret the result geometrically?
Ans.
Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
So,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE...(i)
NowNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE...(ii)
From eq. (i) and (ii) we get
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence proved.
Geometrical Interpretation
According to figure, we have
Area of parallelogram ABCD is

NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Since, the parallelograms on the same base and between the same parallel lines are equal in area
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.11. Find the sine of the angle between the vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE andNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
We know thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.12. If A, B, C, D are the points with position vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEErespectively, find the projection ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
Here, Position vector of A =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Position vector of B =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Position vector of C =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Position vector of D =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Projection ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required projection = √21 .

Q.13. Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).
Ans.
Given that A(1, 2, 3), B(2, –1, 4) and C(4, 5, –1)
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Area of ΔABC =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE=NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
=NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required area isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.14. Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.
Ans.
Let ABCD and ABFE be two parallelograms on the same base AB and between same parallel lines AB and DF.
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEELetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ Area of parallelogram ABCD =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Now Area of parallelogram ABFE =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence proved.


LONG ANSWER TYPE QUESTIONS

Q.15. Prove that in any triangle ABC,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEwhere a, b, c are the magnitudes of the sides opposite to the vertices A, B, C, respectively.
Ans.
Here, in the given figure, the components of c are c cos A and c sin A.
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
In DBDC,
a2 = CD2 + BD2
⇒ a2 = (b – c cos A)+ (c sin A)2
⇒ a2 = b2 + c2 cos2 A – 2bc cos A + c2 sin2 A
⇒ a2 = b2 + c2(cos2 A + sin2 A) – 2bc cos A
⇒ a2 = b2 + c2 – 2bc cos A
⇒ 2bc cos A = b2 + c2 - a2
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence Proved.

Q.16. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEdetermine the vertices of a triangle, show that
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEgives the vector area of the triangle. Hence deduce the condition that the three pointsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE are collinear. Also find the unit vector normal to the plane of the triangle.
Ans.
Since,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare the vertices of ΔABC
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
For three vectors are collinear, area of ΔABC = 0
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
which is the condition of collinearity ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEbe th e unit vector normal to the plane of the ΔABC
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.17. Show that area of the parallelogram whose diagonals are given by NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEisNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE. Also find the area of the parallelogram whose diagonals are NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEandNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
Let ABCD be a parallelogram such that,
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ by law of triangle, we get
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Adding eq. (i) and (ii) we get,
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Subtracting eq. (ii) from eq. (i) we get
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
So, the area of the parallelogram ABCD =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Now area of parallelogram whose diagonals areNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEandNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required area isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.18. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEfind a vector NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE such that  NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Also given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Since,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

On comparing the like terms, we get
c3 – c2 = 0 ...(i)
c1 – c3 = 1 ...(ii)

and c2 – c1 = –1 ...(iii)
Now
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

∴ c1 + c2 + c3 = 3 ...(iv)
Adding eq. (ii) and eq. (iii) we get,
c2 – c3 = 0 ...(v)
From (iv) and (v) we get
c1 + 2c2 = 3 ...(vi)
From (iii) and (vi) we get
Adding
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
c3 – c2 = 0
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Now c2 – c1 = – 1 ⇒NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE


OBJECTIVE TYPE QUESTIONS

Q.19. The vector in the direction of the vectorNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthat has magnitude 9 is
(a)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(b)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(c)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(d)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans. (c)
Solution.
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Unit vector in the direction ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ Vector of magnitude 9 =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (c).

Q.20. The position vector of the point which divides the join of points
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEin the ratio 3 : 1 is
(a)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(b)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(c)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(d)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans. (d)
Solution.
The given vectors areNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEand the ratio is 3 : 1.
∴ The position vector of the required point c which divides the join of the given vectors NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (d).

Q.21. The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is
(a)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(b)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(c)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(d)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans. (c)
Solution.
Let A and B be two points whose coordinates are given as (2, 5, 0) and (– 3, 7, 4)
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (c).

Q.22. The angle between two vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEwith magnitudes √3 and 4, respectively, andNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(a) π/6
(b) π/3
(c) π/2
(d) 5π/2
Ans. (b)
Solution.
Here, given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ From scalar product, we know that
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (b).

Q.23. Find the value of λ such that the vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

are orthogonal
(a) 0 
(b) 1 
(c) 3/2
(d) -5/2
Ans. (d)
Solution.
Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
SinceNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare orthogonal  
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
⇒ 2 + 2λ + 3 = 0
⇒ 5 + 2λ = 0 ⇒NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (d).

Q.24. The value of λ for which the vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
are parallel is
(a) 2/3
(b) 2/3
(c) 5/2
(d) 2/5
Ans. (a)
Solution.

Let
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Since the given vectors are parallel,
∴ Angle between them is 
soNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Squaring both sides, we get
900 + λ2 + 60λ = 46(20 + λ2)
⇒ 900 + λ2 + 60λ = 920 + 46λ2
⇒ λ2 – 46λ2 + 60λ + 900 – 920 = 0
⇒ - 45λ2 + 60λ - 20 = 0
⇒ 9λ2 – 12λ + 4 = 0
⇒(3λ – 2)2 = 0
⇒3λ – 2 = 0
⇒ 3λ = 2
∴ λ = 2/3
Alternate method:
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (a).

Q.25. The vectors from origin to the points A and B are
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE,respectively, then the area of triangle OAB is   
(a) 340 
(b) √25 
(c) √229 
(d)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans. (d)
Solution.

Let O be the origin
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
andNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
∴ Area of ΔOAB =NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence the correct option is (d).

Q.26. For any vectorNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthe value ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis equal to
(a)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(b)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(c)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(d)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans. (d)
Solution.

LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Now,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
SimilarlyNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
andNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (d).

Q.27. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthen value ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis
(a) 5 

(b) 10 
(c) 14 
(d) 16
Ans. (d)
Solution.

Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

⇒ 12 = 10 × 2 × cos θ
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NowNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (d).

Q.28. The vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare coplanar if
(a) λ = –2 
(b) λ = 0 
(c) λ = 1 
(d) λ = – 1
Ans. (a)
Solution.

LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare coplanar, then
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

⇒ l(λ2 – 1) – 1 (λ + 2) + 2(–1 – 2l) = 0
⇒ λ3 – λ – λ – 2 – 2 – 4λ = 0
⇒ λ3 – 6λ – 4 = 0
⇒ (λ + 2) (λ2 – 2λ – 2) = 0
⇒ λ = – 2 or λ2 – 2λ – 2 = 0
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (a).

Q.29. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare unit vectors such thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthen the value of
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

(a) 1
(b) 3
(c) -3/2

(d) None of these
Ans. (c)
Solution.

Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
andNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (c).

Q.30. Projection vector ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis
(a)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(b)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(c)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(d)NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans. (a)
Solution.

The projection vector ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (a).

Q.31. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare three vectors such thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
then value ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(a) 0 
(b) 1 
(c) – 19 
(d) 38
Ans. (c)
Solution.

Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE    
andNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (c).

Q.32. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEand −3 ≤ λ ≤ 2 , then the range ofNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis   
(a) [0, 8]
(b) [– 12, 8]     
(c) [0, 12]     
(d) [8, 12]

Ans. (b)
Solution.

Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NowNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Here - 3 ≤ λ ≤ 2
⇒ - 3.4 ≤ 4λ  ≤ 2.4
⇒ - 12 ≤ 4λ  ≤ 8
∴ 4λ  = [- 12, 8]
Hence, the correct option is (b).

Q.33. The number of vectors of unit length perpendicular to the vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEandNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
(a) one 
(b) two 
(c) three 
(d) infinite
Ans. (b)
Solution.

The number of vectors of unit length perpendicular to vectors
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
So, there will be two vectors of unit length perpendicular to vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the correct option is (b).


FILL IN THE BLANKS

Q.34. The vectorNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEbisects the angle between the non-collinear vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEif ________.
Ans.
If vectorNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE bisects the angle between non-collinear vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE 
then the angle betweenNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis equal to the angle betweenNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
So,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE  ...(i)

Also,NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE[∵ θ is same]   ...(ii)
From eq. (i) and eq. (ii) we get,
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required filler isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.35. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEfor some non-zero vectorNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthen the value of NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis   ________
Ans.
IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis a non-zero vector, thenNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEcan be in the same plane.
Since angles between NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE and  NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE  are zero i.e. θ = 0
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence the required value is 0.

Q.36. The vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEa re the adjacent sides of a parallelogram. The acute angle between its diagonals is ________.
Ans.
Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
andNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Let θ be the angle between the two diagonal vectorsNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEENCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
then
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence the value of required filler isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.37. The values of k for whichNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE is parallel toNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEholds true are _______.
Ans.
Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Now sinceNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis parallel toNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Here we see that atNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEbecome null vector and then it will not be
parallel toNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Now sinceNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis parallel toNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Here we see that atNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEbecome null vector and then it will not be
parallel toNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE 
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the required value of k ∈ (- 1, 1) and k ≠NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.38. The value of the expressionNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis _______.
Ans.
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the value of the filler isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

Q.39. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEandNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis equal to _______.
Ans.
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the value of the filler is 3.

Q.40. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEany non-zero vector, thenNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
equals _______.
Ans.
LetNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
= a1
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the value of the filler isNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE

State True or False in each of the following Exercises.
Q.41. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthen necessarily it impliesNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthenNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEwhich is true.
Hence, the statement is True.

Q.42. Position vector of a point P is a vector whose initial point is origin.

Ans.
True

Q.43. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEthen the vectors
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare orthogonal.
Ans.
Given thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Squaring both sides, we get
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
which implies thatNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare orthogonal.
Hence the given statement is True.

Q.44. The formulaNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEis valid for non-zero vectors
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Hence, the given statement is False.

Q.45. IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEEare adjacent sides of a rhombus, thenNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
Ans.
IfNCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE
So the angle between the adjacent sides of the rhombus should be 90° which is not possible.
Hence, the given statement is False.

The document NCERT Exemplar: Vectors | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on NCERT Exemplar: Vectors - Mathematics (Maths) Class 12 - JEE

1. What is a vector?
Ans. A vector is a mathematical quantity that has both magnitude and direction. It is represented by an arrow, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction of the vector.
2. How is the magnitude of a vector calculated?
Ans. The magnitude of a vector is calculated by using the Pythagorean theorem. It is the square root of the sum of the squares of its components. For example, if a vector has components (a, b, c), then its magnitude is given by √(a^2 + b^2 + c^2).
3. What is the difference between a scalar and a vector quantity?
Ans. A scalar quantity only has magnitude and no direction, while a vector quantity has both magnitude and direction. Examples of scalar quantities include temperature and mass, while examples of vector quantities include velocity and force.
4. How are vectors represented mathematically?
Ans. Vectors can be represented mathematically using various notations. One common notation is to write the components of the vector as a column or row matrix. For example, a vector A = (a1, a2, a3) can be represented as a column matrix [a1, a2, a3] or as a row matrix [a1, a2, a3]T, where T represents the transpose of the matrix.
5. What is the dot product of two vectors?
Ans. The dot product of two vectors is a scalar quantity that is calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them. It is denoted by a dot (·) between the two vectors. The dot product can be used to determine the angle between two vectors or to calculate the work done by a force.
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