Vector Algebra Class 12th Mathematics Part Ii CBSE Solution

Class 12^th Mathematics Part Ii CBSE Solution

Exercise 10.1

1. Represent graphically a displacement of 40 km, 30° east of north.…
2. Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 meters…
3. Classify the following as scalar and vector quantities. (i) time period (ii)…
4. In Fig 10.6 (a square), identify the following vectors. (i) Coinitial (ii)…
5. Answer the following as true or false. (i) vector a and - vector a are…

Exercise 10.2

1. Compute the magnitude of the following vectors: vector a = i + j+k vector b = 2…
2. Write two different vectors having same magnitude.
3. Write two different vectors having same direction.
4. Find the values of x and y so that the vectors 2 i+3 j x i+y j are equal.…
5. Find the scalar and vector components of the vector with initial point (2, 1)…
6. Find the sum of the vectors vector a = i-2 j + k , vector b = - 2 i+4 j+5 k ,…
7. Find the unit vector in the direction of the vector vector a = i + j+2 k…
8. Find the unit vector in the direction of vector vector pq where P and Q are the…
9. For given vectors, vector a = 2 i - j+2 k , vector b = - i + j - k find the…
10. Find a vector in the direction of vector 5 i - j+2 k which has magnitude 8…
11. Show that the vectors 2 i-3 j+4 k , - 4 i+6 j-8 k are collinear.
12. Find the direction cosines of the vector i+2 j+3 k
13. Find the direction cosines of the vector joining the points A(1, 2, -3) and…
14. Show that the vector i + j + k is equally inclined to the axes OX, OY and OZ.…
15. Find the position vector of a point R which divides the line joining two…
16. Find the position vector of the mid point of the vector joining the points…
17. Show that the points A, B and C with position vectors, vector a = 3 i-4 j-4 k…
18. In triangle ABC (Fig 10.18), which of the following is not true: (a) (b) (c)…
19. vector a vector b are two collinear vectors, then which of the following are…

Exercise 10.3

1. Find the angle between two vectors vector a vector b with magnitudes √3 and 2…
2. Find the angle between the vectors i-2 j+3 k 3 i-2 j + k
3. Find the projection of the vector i - j on the vector i + j
4. Find the projection of the vector i + 3 mathfrakj + 7 k on the vector 7 i - j+8…
5. Show that each of the given three vectors is a unit vector: Also, show that…
6. Find | vector a| | vector b| if (vector a + vector b) , (vector a - vector b) =…
7. Evaluate the product (3 vector a-5 vector b) (2 vector a+7 vector b)…
8. Find the magnitude of two vectors vector a vector b , having the same magnitude…
9. Find | vector x| if for a unit vector vector a , (vector x - vector a) (vector…
10. If vector a = 2 i-2 j+3 k , vector b = - i+2 j + k vector c = 3 i + j are such…
11. Show that | vector a| vector b+| vector b| vector a is perpendicular to |…
12. If vector a vector a = 0 and vector a vector b = 0 then what can be concluded…
13. If vector a , vector b , vector c are unit vectors such that vector a + vector…
14. If either vector vector a = vector 0 vector b = vector 0 then vector a vector…
15. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (-1, 0, 0), (0, 1,…
16. Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.…
17. Show that the vectors 2 i - j + k , i-3 j-5 k 3 i-4 j-4 k form the vertices of…
18. If vector a is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then…

Exercise 10.4

1. Find | vector a x vector b| if vector a = i - j j+7 k vector b = 3 i-2 j+2 k…
2. Find a unit vector perpendicular to each of the vector
3. If a unit vector vector a makes angles pi /3 i , pi /4 j and an acute angle θ…
4. Show that (vector a - vector b) x (vector a + vector b) = 2 (vector a x vector…
5. Find λ and μ if (2 i+6 j+27 k) x (i + lambda j + μ k) = vector 0
6. Given that vector a vector b = 0 vector a x vector b = vector 0 What can you…
7. Let the vectors vector a , vector b , vector c be given as Then show that…
8. If either vector a = vector 0 vector b = vector 0 then vector a x vector b =…
9. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5,…
10. Find the area of the parallelogram whose adjacent sides are determined by the…
11. Let the vectors vector a vector b be such that | vector a| = 3 | vector b| =…
12. Area of a rectangle having vertices A, B, C and D with position vectors - i +…

Miscellaneous Exercise

1. Write down a unit vector in XY-plane, making an angle of 30° with the positive…
2. Find the scalar components and magnitude of the vector joining the points P(x1,…
3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of…
4. If | vector a| = | vector b|+| vector c| , then is it true that | vector a| = |…
5. Find the value of x for which x (i + j + k) is a unit vector.
6. Find a vector of magnitude 5 units, and parallel to the resultant of the…
7. If vector a = i + j + k , vector b = 2 i - j+3 k , and vector c = i-2 j + k ,…
8. Show that the points A(1, - 2, - 8), B(5, 0, -2) and C(11, 3, 7) are collinear,…
9. Find the position vector of a point R which divides the line joining two points…
10. The two adjacent sides of a parallelogram are 2 i-4 j+5 k and i-2 j-3 k . Find…
11. Show that the direction cosines of a vector equally inclined to the axes OX,…
12. Let vector a = i + 4 j + 2 k , vector b = 3 i-2 j+7 k and vector c = 2 i - j+4…
13. The scalar product of the vector i + j + k a unit vector along the sum of…
14. If vector a , vector b , vector c are mutually perpendicular vectors of equal…
15. Prove that (vector a + vector b) (vector a + vector b) = | vector a|^2 + |…
16. If theta is the angle between two vectors vector a and vector b , then vector…
17. Let vector a and vector b be two unit vectors and q is the angle between them.…
18. The value of isA. 0 B. -1 C. 1 D. 3
19. If θ is the angle between any two vectors vector a and vector b , then |…

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Exercise 10.1

Question 1.

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

Answer:

North (N), South (S), East (E), and West (W) are plotted on the paper as shown.

Let  be the displacement vector such that  = 40 km

Vector  makes an angle of 30° east of North i.e. 30° with North in North-East direction.

Question 2.

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 meters north-west

(iii) 40° (iv) 40 watt

(v) 10^-19 coulomb (vi) 20 m/s²

Answer:

Scalars are defined as quantities that have magnitude only.

Vectors are defined as a quantity that has magnitude as well as direction.

(i) 10 kg: It is a measure of mass. It is a scalar quantity as it has magnitude only and no direction.

(ii) 2 meters north-west: It is a measure of distance in a particular direction. ∴ It is a vector quantity as it has magnitude as well as direction.

(iii) 40°: It is a measure of angle. It is a scalar quantity as it has magnitude only and no direction.

(iv) 40 Watt: It is a measure of power. It is a scalar quantity as it has magnitude only and no direction.

(v) 10^-19 coulomb: It is a measure of electric charge. It is a scalar quantity as it has magnitude only and no direction.

(vi) 20 m/sec²: It is a measure of acceleration. It is a vector quantity as it is a measure of rate of change of velocity.

Question 3.

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

Answer:

Scalars are defined as quantities that have magnitude only.

Vectors are defined as a quantity that has magnitude as well as direction.

(i) time period: It is a scalar quantity as it has magnitude only and no direction.

(ii) distance: It is a scalar quantity as it has magnitude only and no direction.

(iii) force: It is a vector quantity as it has magnitude as well as direction.

(iv) velocity: It is a vector quantity as it has magnitude as well as direction.

(v) work done: It is a scalar quantity as it has magnitude as well as direction.

Question 4.

In Fig 10.6 (a square), identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

Answer:

(i) Coinitial vectors: Two or more vectors having same initial point are called coinitial vectors.

So, in the above figure  and  are coinitial vectors as they have same initial point.

∴ Coinitial vectors:  and 

(ii) Equal vectors: Two or more vectors having same direction and same magnitude are called equal vectors.

So, in the above figure  and  are equal vectors as they have same magnitude and same direction.

∴ Equal vectors:  and 

(iii) Collinear but not equal:

∵  and  are parallel vectors, so, they are collinear. But they have opposite direction, so, they are not equal.

Hence,  and  are collinear but not equal.

Question 5.

Answer the following as true or false.

(i)  and  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)  and  are collinear: True

Explanation:  and  are parallel vectors, so, they are collinear.

(ii) Two collinear vectors are always equal in magnitude: False

Explanation: we know,  and  are unit vectors, so they have same magnitude but  and  are vectors along x – axis and y – axis respectively.

(iii) Two vectors having same magnitude are collinear: False

Explanation: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

We know,  and  are unit vectors, so they have same magnitude but  and  are vectors along x – axis and y – axis respectively. Since, they are not parallel to each other, so, they are not collinear.

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

Explanation: Two vector are said to be equal, if they have the same magnitude and direction.

 and  are collinear vectors as they are parallel to each other and also they have same magnitude. But they do not have same direction, so, they are unequal vectors.

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Exercise 10.2

Question 1.

Compute the magnitude of the following vectors:

Answer:

Question 2.

Write two different vectors having same magnitude.

Answer:

Let 

And

We can see clearly  because the all the coefficients of and  are not same. In  and  , the coefficients are different. Now, we check the magnitude of both,

And

So, magnitude of both vectors is same but they are different.

Question 3.

Write two different vectors having same direction.

Answer:

Let

And

So, here  where m = 2 > 0

Therefore, both vectors have same direction.

Now, we check for the magnitude, 

So, they have same direction but same magnitude.

Question 4.

Find the values of x and y so that the vectors  are equal.

Answer:

For two vectors to be equal, the coefficients of both vectors should be equal.

Comparing the -coefficient, we get x=2

Comparing the -coefficient, we get y=3

Question 5.

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

Answer:

Let A be the initial point (2,1) and B be the final point (-5,7).

So,  and 

Now we want to find the 

Therefore, the scalar components of  are -7 and 6. The vector components of  are .

Question 6.

Find the sum of the vectors 

Answer:

And

We want to find 

To find the sum, we add coefficients of  to each other.

So,

Question 7.

Find the unit vector in the direction of the vector 

Answer:

We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as 

So, we find the magnitude of  first.

Question 8.

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:

So,  and 

Now we want to find the 

Now, we have to find the unit direction in the direction of .We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as 

So, we find the magnitude of  first.

Question 9.

For given vectors,  find the unit vector in the direction of the vector 

Answer:

We want to find 

To find the sum, we add coefficients of  to each other.

So,

Now, we have to find the unit direction in the direction of.We know that unit vector means that the magnitude of the vector is 1(unit).

So, we find the magnitude of  first.

Question 10.

Find a vector in the direction of vector which has magnitude 8 units.

Answer:

Let 

The vector in the direction of  having unit magnitude is .

So, The vector in the direction of  having magnitude 8 units =.

Question 11.

Show that the vectors  are collinear.

Answer:

We know that two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form  where a and b are vectors and 'm' is a scalar quantity.

From the question,

Let  and 

Here,

So, where, m=-2.

∴ The given two vectors are collinear.

Question 12.

Find the direction cosines of the vector 

Answer:

The direction cosines of a vector are defined as the coefficients of  in the unit vector in the direction of the vector.

So, first we find the unit vector in the direction of the vector.

Let 

Therefore, The direction cosines of the given vector are .

Question 13.

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:

So,  and 

Now we want to find the 

Now, we have to find the direction cosines which are the coefficients of the unit vector in the direction of .We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as 

So, we find the magnitude of  first.

Therefore, The direction cosines of the given vector are .

Question 14.

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

Answer:

To find the inclination of the vector with OX, OY, OZ. We find the direction cosines of the vector.

We know that the direction cosines of a vector are defined as the coefficients of  in the unit vector in the direction of the vector.

So, first we find the unit vector in the direction of the vector.

Let 

Therefore, The direction cosines of the given vector are .

Let  be the angle between  and OX.

Therefore, 

Similarly, Let  be the angle between  and OY.

Therefore, 

And, be the angle between  and OZ.

Therefore, 

Therefore, 

Hence proved that the vector is equally inclined with the axes OX, OY, and OZ.

Question 15.

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 ratio 2 : 1

(i) internally (ii) externally

Answer:

Given that  and 

(i) The  lies on the segment PQ(internal division).If m:n is the ratio in which  divides PQ, then

Given m:n=2:1, m=2 and n=1

(ii) The  does not lie on the segment PQ(external division).If m:n is the ratio in which  divides PQ,then

Given m:n=2:1, m=2 and n=1

Question 16.

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

Answer:

So,  and 

Let  be the mid point of .

Question 17.

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

Answer:

O be the origin,
Let 

And

Now we find the vectors 

 ……….(1)

 ………….(2)

 ……….(3)

Adding equations (1), (2) and (3)

Therefore, ABC form a triangle.

Now, we want to prove that it is a right angled triangle.

Therefore, the triangle satisfies the Pythagoras theorem. Hence, proved the given vectors form a right angled triangle.

Question 18.

In triangle ABC (Fig 10.18), which of the following is not true:



(a) 

(b) 

(c) 

(d) 

Answer:

We know by triangle law of vectors,

 …………..(1)

Therefore, 

Hence (B) is true.

We know that 

Putting in eq(1), we get

 …….(2)

Hence (A) is also true.

Now, 

Putting in eq(2)

Hence (D) is correct.

We are asked the option which is not true.

Therefore, the correct option is (C).

Question 19.

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

(a) 

(b) 

(c) the respective components of  are not proportional

(d) both the vectors  have same direction, but different magnitudes.

Answer:

We know that two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form  where a and b are vectors and ' m ' is a scalar quantity.

Therefore, (a) is true.

In (b), 

So, (b) is also true.

The vectors  and  are proportional,
Therefore, (c) is not true.

The vectors  and  can have different magnitude as well as different direction.

Therefore, (d) is not true.

We are asked the options which are not true.

∴ The correct answer is (c) and (d).

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Exercise 10.3

Question 1.

Find the angle between two vectors  with magnitudes √3 and 2 respectively having 

Answer:

Given,

We know, 

Putting the value of 

∴ 

So, the angle between the vectors  is π/4

Question 2.

Find the angle between the vectors 

Answer:

Given vectors are 

Let 

 =

∴  We know,

⇒ 

⇒ 

⇒ cos θ = 10/14

⇒ θ = cos^-1(5/7)

Question 3.

Find the projection of the vector  on the vector 

Answer:

Let  and 

Projection of  is given by 

Now 

∴ Projection of 

= 0

∴ Projection of  is 0.

Question 4.

Find the projection of the vector  on the vector 

Answer:

Let 

Projection of  is given by 

Now 

∴ Projection of 

∴ Projection of .

Question 5.

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

Also, show that they are mutually perpendicular to each other.

Answer:

Let 

Now we need to find out the magnitude of 

Since, 

∴ the three given vectors are unit vectors.

To show that each of three vectors are mutually perpendicular to each other

We have to show 

Since, 

∴  are mutually perpendicular to each other.

Question 6.

Find if 

Answer:

Given, 

Now, 

∴ 

⇒ 

∴ 

Question 7.

Evaluate the product 

Answer:

To find 

Question 8.

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

Answer:

Let θ be the angle between 

Given, 

 and θ = 60°

Scalar product of 

i.e 

We know 

∴  [magnitude of vector is positive]

So, 

Hence, 

Question 9.

Find  if for a unit vector 

Answer:

Given, vector  is a unit vector

∴ 

∴ 

Question 10.

If  are such that  is perpendicular to  then find the value of λ.

Answer:

Given, 

It is given  is perpendicular to , then

∴ λ = 8

Question 11.

Show that  is perpendicular to  for any two nonzero vectors .

Answer:

To show  is perpendicular to  for any two non zero vectors , we need to show dot product of  is zero.

= 0

Since, dot product of  is zero.

∴  is perpendicular to 

Question 12.

If  and  then what can be concluded about the vector 

Answer:

Given, 

⇒ 

⇒ 

∴  is a zero vector.

Hence, can be any vector.

Question 13.

If  are unit vectors such that  find the value of 

Answer:

Given  are unit vectors and 

Now

Adding (i), (ii) and (iii) we get

∴ 

Question 14.

If either vector  then  But the converse need not be true. Justify your answer with an example.

Answer:

If either  = 0

Now let 

So, 

∴ 

∴ 

Since 

Hence, the converse of given statement need not be true.

Question 15.

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC, [∠ABC is the angle between the vectors ].

Answer:

Given points are A (1, 2, 3), B (-1, 0, 0) and C (0, 1, 2)

∠ABC is the angle between vectors 

We know

⇒ ∠ABC = cos^-1(10/√102)

Question 16.

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

Answer:

Given points are A (1, 2, 7), B (2, 6, 3) and C (3, 10, -1)

So,

Since, 

∴ The given points A, B and C are collinear.

Question 17.

Show that the vectors  form the vertices of a right angled triangle.

Answer:

Let vectors  be position vectors of points A, B and C respectively.

Then, 

Now vectors  represent sides of ΔABC

Now 

And

∴ 

From Pythagoras theorem we know

If  then ΔABC is a right-angled triangle

∴ Δ ABC is a right-angled triangle.

Question 18.

If  is a 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:

Given λ is a unit vector.

And 

Since 

∴ 

λ is a unit vector if a = 1/|λ|

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Exercise 10.4

Question 1.

Find  if 

Answer:

Given that
 and 

Expanding along first row,

Question 2.

Find a unit vector perpendicular to each of the vector 

Answer:

Given that
 and

Let 

Let 

Now, we want to find a vector which is perpendicular to both  and . It is given by .

Let 

Expanding along first row,

Therefore, 

Now we want to find the unit vector. We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as 

So, we find the magnitude of  first.

Question 3.

If a unit vector  makes angles  and an acute angle θ with then find θ and hence, the components of .

Answer:

Let 

Given that it is a unit vector,

So,  ……(1)

Let  be the angle  with  respectively.

Then,

 (given)

 (given)

Putting value of x,y, z in equation (1)

 (Squaring both sides)

As  should be acute, 

Components of  are the coefficients of , which are,

Question 4.

Show that 

Answer:

We solve for the left-hand side,

L.H.S.=

We know  and 

Also, 

Putting these in our equation, we get

which is equal to R.H.S.

Hence proved.

Question 5.

Find λ and μ if 

Answer:

Let
 and 

Given that 

Expanding along first row,

By comparing the y-coefficients,

By comparing the z-coefficients,

These values should also satisfy the equation we will get from comparing the x-coefficients,

Therefore, .

Question 6.

Given that  What can you conclude about the vectors ?

Answer:

Given that

 (where θ is the angle between the vectors)

Also given that,

 (where θ is the angle between the vectors)

As there is no value of θ for which both sin θ and cos θ are zero.

Therefore, the condition for which  and  is:

.

Question 7.

Let the vectors  be given as  Then show that 

Answer:

Given that

and

Solving for left hand side,

First, we calculate 

 (Property of determinant)

Which is equal to R.H.S.

Hence proved.

Question 8.

If either  then  Is the converse true? Justify your answer with an example.

Answer:

 (where θ is the angle between the vectors)

If 

Similarly, If 

Now, If 

 (where θ is the angle between the vectors)

sin θ = 0, this implies θ = 0°

This implies that the converse is not always true. The vectors may not be zero but the angle between is 0°, i.e. the vectors are parallel.

Question 9.

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

Answer:

So, 

Now we want to find the 

Now we want to find the 

We know that 

So, we find  first.

Expanding along first row,

Question 10.

Find the area of the parallelogram whose adjacent sides are determined by the vectors 

Answer:

Given
 and

We know that 

So, we find ,

Expanding along first row,

Question 11.

Let the vectors  be such that  then  is a unit vector, if the angle between  is
A. π/6

B. π/4

C. π/3

D. π/2

Answer:

Given that:

 (Magnitude of unit vector is 1)

Putting the values in the equation (1), we get

 (where θ is the angle between the vectors)

Therefore, The correct option is (B).

Question 12.

Area of a rectangle having vertices A, B, C and D with position vectors respectively is
A. 1/2

B. 1

C. 2

D. 4

Answer:

Let

Now we want to find the 

Now we want to find the 

We know that 

So, we find ,

Expanding along first row,

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Miscellaneous Exercise

Question 1.

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

Answer:

Given: A unit vector in XY- plane.

Let  is a unit vector in the given XY- plane then the value of 

Will be : 

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

∴ for θ = 30°

Hence, the required unit vector is .

Question 2.

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

Answer:

Given: points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are given.

The vector obtained by joining the given points P and Q:

 = position vector of Q – position vector of P

Hence the scalar component of the vector obtained by joining the points are

[(x₂-x₁), (y₂-y₁), (z₂-z₁)]

And the magnitude of the vector obtained by joining the points is

Question 3.

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 be the initial position and B be the final position of the girl.

Position of girl will be as shown in figure:

Hence,  and ∠ BAO = 60°

Now, by using triangle law of vector addition,

Hence girls displacement from initial to final position is : 

Question 4.

If , then is it true that ? Justify your answer.

Answer:

Let in given triangle 

Now, by using triangle law of vector addition,

As we can see that represent the sides of triangle.

Also we know that sum of two sides of a triangle must be greater than its third side.

 is not true.

Question 5.

Find the value of x for which  is a unit vector.

Answer:

Given:  as a unit vector.

Now, if is a unit vector then

Question 6.

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors  and 

Answer:

Given: 

Let the resultant of 

Then 

Then

Now the vector of magnitude 5 units and parallel to  is:

Question 7.

If , and , find a unit vector parallel to the vector .

Answer:

Given: 

Let the resultant of 

Then 

Then

∴ unit vector along

Question 8.

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:

Given: A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)

Then

Then

Thus the given points are collinear.

Now to find the ratio in which B divides AC. Let it be λ :1

On equating the terms, we get:

5(λ + 1) = 11λ + 1

⇒ 5λ + 5 = 11λ + 1

⇒ 4 = 6λ

⇒ λ = 4/6 = 2/3

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

Question 9.

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 midpoint of the line segment RQ.

Answer:

Given: points  are given.

Point R is given which divides P and Q in the ratio 1:2.

Then

∴ position vector of R is 

And position vector of mid-point of RQ = 

Hence, P is mid-point of the line segment RQ.

Question 10.

The two adjacent sides of a parallelogram are  and . Find the unit vector parallel to its diagonal. Also, find its area.

Answer:

Given: Two adjacent sides of a parallelogram are 

Then the diagonal of parallelogram is given by the resultant of .

Let the diagonal is .

Then 

Then

∴ unit vector parallel to its diagonal is 

And area of parallelogram ABCD is 

∴ 

Hence, area of parallelogram ABCD is .

Question 11.

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

.

Answer:

let the vector equally inclined to OX, OY and OZ at angle α.

Then the direction cosine of the vectors are cos α, cos α and cos α.

Because,

Cos²α + Cos²α + Cos²α = 1

⇒ 3 Cos²α = 1

⇒ Cos²α = 1/3

Hence, the direction cosines of the vector which are equally inclined to the axis are

.

Question 12.

Let  and . Find a vector  which is perpendicular to both  a and , and 

Answer:

Given: 

Let 

Because  is perpendicular to both  and .

Then 

d₁ + 4d₂ + 2d₃ = 0 .......(1)

And 

3d₁ - 2d₂ + 7d₃ = 0 ......(2)

And  (given)

2d₁ - d₂ + 4d₃ = 15.......(3)

So we have the equations to solve as,
d₁ + 4d₂ + 2d₃ = 0
3d₁ - 2d₂ + 7d₃ = 0
2d₁ - d₂ + 4d₃ = 15

From equation (1),
d₁ = - 4d₂ - 2d₃ ........(4)

Putting this value in equation 2 we get,

-12d₂ - 6d₃ - 2d₂ + 7d₃ = 0

Therefore,
14d₂ = d₃ .......(5)

Now putting the value of d₃ in equation 4 we get,
d₁ = -4d₂ - 28d₂
d₁ = -32d₂........(6)

Putting (5) and (6) in equation (3) we get,

-64d₂ - d₂ + 56d₂ = 15
-9d₂ = 15
d₂ = -5/3

Now we can find other values as well,

Hence 

Hence, the required vector is 

Question 13.

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

Answer:

Given: given vectors are 

Then sum of vector is given by the resultant of .

Let the sum is .

Then 

Then magnitude of  is :

Scalar product of 

Square on both sides:

⇒ (λ + 6)² = (λ)² + 4λ + 44

⇒ (λ)² + 12λ + 36 =(λ)² + 4λ + 44

⇒ 8λ = 8

⇒ λ = 1

Question 14.

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

Answer:

Given: vectors are mutually perpendicular to each other and are of equal magnitude.

And 

Let the vector  be inclined to  at angles α, β and γ respectively.

Then, we have

From (1), (2) and (3)

As 

Hence, 

Hence, the vector  are equal inclined to .

Question 15.

Prove that , if and only if are perpendicular, given .

Answer:

given: (

To prove: vectors are mutually perpendicular to each other.

Hence, are mutually perpendicular to each other.as  is given.

Question 16.

If  is the angle between two vectors and , then  only when
A.

B. 

C. 0 < θ < π

D. 0 ≤ θ < π

Answer:

let θ is the angle between two vectors .

Then are non-zero vectors so that are positive.

As we know 

For 

As  are positive.

Hence, 

The correct answer is (B).

Question 17.

Let and be two unit vectors and q is the angle between them. Then  is a unit vector if
A. 

B. 

C. 

D. 

Answer:

let the two unit vectors are and  is the angle between.

Then 

Then this is () is unit vector if 

scalar product is cumulative.

Hence, () is unit vector if .

The correct answer is (D).

Question 18.

The value of  is
A. 0

B. –1

C. 1

D. 3

Answer:

given: 

=1-1 + 1

=1

The correct answer is (C).

Question 19.

If θ is the angle between any two vectors and , then  when θ is equal to
A. 0

B. π/4

C. π/2

D. π

Answer:

let θ is the angle between two vectors .

Then are non zero vectors so that are positive.

Thus when .

The correct answer is (B).