NCERT Solutions - Vector Algebra JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : NCERT Solutions - Vector Algebra JEE Notes | EduRev

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Question 1: Represent graphically a displacement of 40 km, 30° east of north.

ANSWER : -

 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Here, vector  represents the displacement of 40 km, 30° East of North.

Question 2: Classify the following measures as scalars and vectors.

(i) 10 kg 

(ii) 2 metres north-west 

(iii) 40°

(iv) 40 watt 

(v) 10–19 coulomb

(vi) 20 m/s2

ANSWER : - (i) 10 kg is a scalar quantity because it involves only magnitude.

(ii) 2 meters north-west is a vector quantity as it involves both magnitude and direction.

(iii) 40° is a scalar quantity as it involves only magnitude.

(iv) 40 watts is a scalar quantity as it involves only magnitude.

(v) 10–19 coulomb is a scalar quantity as it involves only magnitude.

(vi) 20 m/s2 is a vector quantity as it involves magnitude as well as direction.

 

Question 3: Classify the following as scalar and vector quantities.

(i) time period                                   

(ii) distance                                       

(iii) force

(iv) velocity                                     

(v) work done

ANSWER : - (i) Time period is a scalar quantity as it involves only magnitude.

(ii) Distance is a scalar quantity as it involves only magnitude.

(iii) Force is a vector quantity as it involves both magnitude and direction.

(iv) Velocity is a vector quantity as it involves both magnitude as well as direction.

(v) Work done is a scalar quantity as it involves only magnitude.

 Question 4: In Figure, identify the following vectors.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER : - (i) Vectors  a and  d are coinitial because they have the same initial point.

(ii) Vectors b and d are equal because they have the same magnitude and direction.

(iii) Vectors a and  c are collinear but not equal. This is because although they are parallel, their directions are not the same.

 

Question 5: Answer the following as true or false.

(i) a and b are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

ANSWER : - (i) True.

Vectors  a and b are parallel to the same line.

(ii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iii) False.

It is not necessary for two vectors having the same magnitude to be parallel to the same line.

(iv) False.

Two vectors are said to be equal if they have the same magnitude and direction, regardless of the positions of their initial points.

 Question 6: Compute the magnitude of the following vectors:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER : - The given vectors are:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 7: Write two different vectors having same magnitude.

Ans:-

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 8: Write two different vectors having same direction.

ANSWER : -

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 9: Find the values of x and y so that the vectors NCERT Solutions - Vector Algebra JEE Notes | EduRev are equal.

ANSWER : -The two vectors will be equal if their corresponding components are equal.

Hence, the required values of x and y are 2 and 3 respectively.

 

Question 10: Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

ANSWER : - The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Hence, the required scalar components are –7 and 6 while the vector components are 

 

Question 11: Find the sum of the vectors   NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 12: Find the unit vector in the direction of the vector  NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER : - The unit vector  in the direction of vector  is given by .

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 13: Find the unit vector in the direction of vector , where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

ANSWER : - The given points are P (1, 2, 3) and Q (4, 5, 6).

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 14: For given vectors,  and  , find the unit vector in the direction of the vector 

Answers:-

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 15: Find a vector in the direction of vector  NCERT Solutions - Vector Algebra JEE Notes | EduRev

Answers

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 16: Show that the vectorsNCERT Solutions - Vector Algebra JEE Notes | EduRev  are collinear.

ANSWER : -

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 17: Find the direction cosines of the vector NCERT Solutions - Vector Algebra JEE Notes | EduRev

Answers 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 18: Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B.

ANSWER : - The given points are A (1, 2, –3) and B (–1, –2, 1).

 NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 19: Show that the vector  is equally inclined to the axes OX, OY, and OZ.

ANSWER : -

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Therefore, the direction cosines of 

Now, let αβ, and γbe the angles formed by  with the positive directions of xy, and z axes.

Then, we have

Hence, the given vector is equally inclined to axes OX, OY, and OZ.

Question 20: Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  NCERT Solutions - Vector Algebra JEE Notes | EduRevrespectively, in the ration 2:1

ANSWER : - The position vector of point R dividing the line segment joining two points

P and Q in the ratio m: n is given by:

        i.  Internally:

    NCERT Solutions - Vector Algebra JEE Notes | EduRev

(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,

NCERT Solutions - Vector Algebra JEE Notes | EduRev

(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 21: Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).

ANSWER : - The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 22: Show that the points A, B and C with position vectors NCERT Solutions - Vector Algebra JEE Notes | EduRev, NCERT Solutions - Vector Algebra JEE Notes | EduRev respectively form the vertices of a right angled triangle. 

ANSWER :

Position vectors of points A, B, and C are respectively given as

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 23: In triangle ABC which of the following is not true:

NCERT Solutions - Vector Algebra JEE Notes | EduRev  NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER : -  On applying the triangle law of addition in the given triangle, we have:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Hence, the equation given in alternative C is incorrect.

The correct answer is C.

Question 24: If  are two collinear vectors, then which of the following are incorrect

NCERT Solutions - Vector Algebra JEE Notes | EduRev

C. the respective components of  are proportional

D. both the vectors  have same direction, but different magnitudes

ANSWER : - If  are two collinear vectors, then they are parallel.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Thus, the respective components of  are proportional.

However, vectors  can have different directions.

Hence, the statement given in D is incorrect.

The correct answer is D.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 26: Find the angle between the vectorsNCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER : - The given vectors are .

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 27: Find the projection of the vector NCERT Solutions - Vector Algebra JEE Notes | EduRev on the vector NCERT Solutions - Vector Algebra JEE Notes | EduRev.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 28: Find the projection of the vector NCERT Solutions - Vector Algebra JEE Notes | EduRev  on the vector  NCERT Solutions - Vector Algebra JEE Notes | EduRev.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 29: Show that each of the given three vectors is a unit vector

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 Question 31: Evaluate the product NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER  

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 32: Find the magnitude of two vectors a and b, having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 35: Show that  NCERT Solutions - Vector Algebra JEE Notes | EduRev is perpendicular to  NCERT Solutions - Vector Algebra JEE Notes | EduRev  ,for any two nonzero vectors a and b.

ANSWER : -

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 36: If , then what can be concluded about the vector ?

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 Question 38: If either vector a = 0, then b = 0. But the converse need not be true. Justify your answer with an example.

ANSWER : -

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 39:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER : - The vertices of ΔABC are given as A (1, 2, 3), B (–1, 0, 0), and C (0, 1, 2).

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 40: Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

ANSWER : - The given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 41: Show that the vectors 2i - j k, i - 3j - 5k and 3i - 4j - 4k  form the vertices of a right angled triangle.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 Question 42: NCERT Solutions - Vector Algebra JEE Notes | EduRev nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then NCERT Solutions - Vector Algebra JEE Notes | EduRev is unit vector if

(A) λ = 1

(B) λ = –1     

(c) a = | λ|

(d) a = 1/| λ|

ANSWER :  Vector NCERT Solutions - Vector Algebra JEE Notes | EduRev is a unit vector if  NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 45: If a unit vector  a makes an angles π/3 with  i, π/3 with  j and an acute angle θ with k, then find θ and hence, the compounds of .

ANSWER : - Let unit vector  have (a1a2a3) components.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Also, it is given that  a makes angles π/3, î , π/4, with j^  , and an acute angle θ with k^
Then, we have:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 50: If either  a = 0 or b = 0, then a × b = 0. Is the converse true? Justify your answer with an example.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 51: Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

ANSWER : - The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and

C (1, 5, 5).

The adjacent sides AB and BC of ΔABC are given as:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 52: Find the area of the parallelogram whose adjacent sides are determined by the vector   .

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 54: Area of a rectangle having vertices A, B, C, and D with position vectors NCERT Solutions - Vector Algebra JEE Notes | EduRev and  NCERT Solutions - Vector Algebra JEE Notes | EduRev respectively is 

(A) 1/2

(B) 1

 (C) 2    

(D) 4

ANSWER : - The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

The adjacent sides  NCERT Solutions - Vector Algebra JEE Notes | EduRev of the given rectangle are given as:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Now, it is known that the area of a parallelogram whose adjacent sides are   NCERT Solutions - Vector Algebra JEE Notes | EduRev

Hence, the area of the given rectangle is  NCERT Solutions - Vector Algebra JEE Notes | EduRev sq. unit

The correct answer is C.

 Question 55: Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

ANSWER : -

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Here, θ is the angle made by the unit vector with the positive direction of the x-axis.

Therefore, for θ = 30°:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 56: Find the scalar components and magnitude of the vector joining the points  P(x1,y1, z1) and Q(x2,y2,z2)

ANSWER : - 

The vector joining the points  P(x1,y1, z1) and Q(x2,y2,z2) can be obtained by,                                                                                                                                  

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Hence, the scalar components and the magnitude of the vector joining the given points are respectively    NCERT Solutions - Vector Algebra JEE Notes | EduRev andNCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 57: A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

ANSWER : - Let O and B be the initial and final positions of the girl respectively.

Then, the girl’s position can be shown as:

Now, we have:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev 

By the triangle law of vector addition, we have:

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 58: If NCERT Solutions - Vector Algebra JEE Notes | EduRev then is it true that  NCERT Solutions - Vector Algebra JEE Notes | EduRev  ? Justify your answer.

Answer: 

Now, by the triangle law of vector addition, we have NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

It is clearly known that   NCERT Solutions - Vector Algebra JEE Notes | EduRev represent the sides of ΔABC 

Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Hence, it is not true that NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 60: Find a vector of magnitude 5 units, and parallel to the resultant of the vectors .

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 61: if NCERT Solutions - Vector Algebra JEE Notes | EduRev find a unit vector parallel to the vectoNCERT Solutions - Vector Algebra JEE Notes | EduRev

Answer: 

We have NCERT Solutions - Vector Algebra JEE Notes | EduRev 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 62: Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

 

ANSWER : - The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).

NCERT Solutions - Vector Algebra JEE Notes | EduRev

On equating the corresponding components, we get:

5(λ + 1) = 11λ +1

λ = 2 : 3

Hence, point B divides AC in the ratio 2 : 3

 

Question 63: Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are NCERT Solutions - Vector Algebra JEE Notes | EduRev externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

ANSWER : - It is given that

NCERT Solutions - Vector Algebra JEE Notes | EduRev

It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 64: The two adjacent sides of a parallelogram are 2i - 4j + 5k and i - 2j - 3k .Find the unit vector parallel to its diagonal. Also, find its area.

ANSWER : - Adjacent sides of a parallelogram are given as: a = 2i - 4j 5k and  b = i - 2j - 3k

Then, the diagonal of a parallelogram is given by a + b .

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 65 : Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are .

ANSWER : - Let a vector be equally inclined to axes OX, OY, and OZ at angle α.

Then, the direction cosines of the vector are cos α, cos α, and cos α.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 67: The scalar product of the vector  NCERT Solutions - Vector Algebra JEE Notes | EduRev  with a unit vector along the sum of vectors    NCERT Solutions - Vector Algebra JEE Notes | EduRev and  NCERT Solutions - Vector Algebra JEE Notes | EduRev is equal to one. Find the value of λ

ANSWER : -  

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Therefore, unit vector along NCERT Solutions - Vector Algebra JEE Notes | EduRev + NCERT Solutions - Vector Algebra JEE Notes | EduRev is given as: 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Scalar product of NCERT Solutions - Vector Algebra JEE Notes | EduRev with this unit vector is 1.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 Question 68: If   NCERT Solutions - Vector Algebra JEE Notes | EduRev are mutually perpendicular vectors of equal magnitudes, show that the vector  NCERT Solutions - Vector Algebra JEE Notes | EduRev is equally inclined to  NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER :  Since NCERT Solutions - Vector Algebra JEE Notes | EduRev are mutually perpendicular vectors, we have

NCERT Solutions - Vector Algebra JEE Notes | EduRev

It is given that:

NCERT Solutions - Vector Algebra JEE Notes | EduRevNCERT Solutions - Vector Algebra JEE Notes | EduRevNCERT Solutions - Vector Algebra JEE Notes | EduRev

Question 70: If θ is the angle between two vectors NCERT Solutions - Vector Algebra JEE Notes | EduRev , then  NCERT Solutions - Vector Algebra JEE Notes | EduRev  only when

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

ANSWER : - Let θ be the angle between two vectors  NCERT Solutions - Vector Algebra JEE Notes | EduRev  are non-zero vectors so that 

It is known that  NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

 Question 71: Let   NCERT Solutions - Vector Algebra JEE Notes | EduRev be two unit vectors andθ is the angle between them. Then  NCERT Solutions - Vector Algebra JEE Notes | EduRev  is a unit vector if

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

Answer:

Let   NCERT Solutions - Vector Algebra JEE Notes | EduRev be two unit vectors andθ is the angle between them

Then 

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

 Question 72: The value of  NCERT Solutions - Vector Algebra JEE Notes | EduRev is

(A) 0  

(B) –1

(C) 1 

(D) 3

Answer

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

Question 73: If θ is the angle between any two vectors   NCERT Solutions - Vector Algebra JEE Notes | EduRev  when θ isequal to

Answer: Let θ be the angle between two vectors   NCERT Solutions - Vector Algebra JEE Notes | EduRev are non-zero vectors, so that  NCERT Solutions - Vector Algebra JEE Notes | EduRev are positive.

NCERT Solutions - Vector Algebra JEE Notes | EduRev

NCERT Solutions - Vector Algebra JEE Notes | EduRev

 

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