NCERT Solutions - Vector Algebra JEE Notes | EduRev

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

ANSWER : -

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.

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:

ANSWER : - The given vectors are:

Question 7: Write two different vectors having same magnitude.

Ans:-

Question 8: Write two different vectors having same direction.

ANSWER : -

Question 9: Find the values of x and y so that the vectors  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,

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

Question 11: Find the sum of the vectors   

Question 12: Find the unit vector in the direction of the vector  

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

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).

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

Answers:-

Question 15: Find a vector in the direction of vector  

Answers

Question 16: Show that the vectors  are collinear.

ANSWER : -

Question 17: Find the direction cosines of the vector 

Answers 

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).

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

ANSWER : -

Therefore, the direction cosines of 

Now, let α, β, and γbe the angles formed by  with the positive directions of x, y, 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  respectively, 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:

(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,

(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

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,

Question 22: Show that the points A, B and C with position vectors ,  respectively form the vertices of a right angled triangle. 

ANSWER :

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

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

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

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

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.

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.

Question 26: Find the angle between the vectors

ANSWER : - The given vectors are .

Question 27: Find the projection of the vector  on the vector .

Question 28: Find the projection of the vector   on the vector  .

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

 Question 31: Evaluate the product 

ANSWER  

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.

Question 35: Show that   is perpendicular to    ,for any two nonzero vectors a and b.

ANSWER : -

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

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

ANSWER : -

Question 39:

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

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).

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

 Question 42:  nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then  is unit vector if

(A) λ = 1

(B) λ = –1     

(c) a = | λ|

(d) a = 1/| λ|

ANSWER :  Vector  is a unit vector if 

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 (a₁, a₂, a₃) components.

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:

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

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:

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

Question 54: Area of a rectangle having vertices A, B, C, and D with position vectors  and   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:

The adjacent sides   of the given rectangle are given as:

Now, it is known that the area of a parallelogram whose adjacent sides are  

Hence, the area of the given rectangle is   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 : -

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

Therefore, for θ = 30°:

Question 56: Find the scalar components and magnitude of the vector joining the points  P(x_1,y_1, z₁₎ and Q(x₂,y₂,z₂)

ANSWER : - 

The vector joining the points  P(x_1,y_1, z₁₎ and Q(x₂,y₂,z₂) can be obtained by,                                                                                                                                  

Hence, the scalar components and the magnitude of the vector joining the given points are respectively     and

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:

By the triangle law of vector addition, we have:

Question 58: If  then is it true that    ? Justify your answer.

Answer: 

Now, by the triangle law of vector addition, we have 

It is clearly known that    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.

Hence, it is not true that 

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

Question 61: if  find a unit vector parallel to the vector 

Answer: 

We have  

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).

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  externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

ANSWER : - It is given that

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

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 .

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.

Question 67: The scalar product of the vector    with a unit vector along the sum of vectors     and   is equal to one. Find the value of λ

ANSWER : -  

Therefore, unit vector along  +  is given as: 

Scalar product of  with this unit vector is 1.

 Question 68: If    are mutually perpendicular vectors of equal magnitudes, show that the vector   is equally inclined to 

ANSWER :  Since  are mutually perpendicular vectors, we have

It is given that:

Question 70: If θ is the angle between two vectors  , then    only when

ANSWER : - Let θ be the angle between two vectors    are non-zero vectors so that 

It is known that 

 Question 71: Let    be two unit vectors andθ is the angle between them. Then    is a unit vector if

Answer:

Let    be two unit vectors andθ is the angle between them

Then 

 Question 72: The value of   is

(A) 0  

(B) –1

(C) 1 

(D) 3

Answer

Question 73: If θ is the angle between any two vectors     when θ isequal to

Answer: Let θ be the angle between two vectors    are non-zero vectors, so that   are positive.