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## Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra

### Plus Two Maths Vector Algebra Three Mark Questions and Answers

Question 1.

Find \(\bar{a}+\bar{b}, \bar{a}-\bar{b}\) and \(\bar{b}+\bar{c}\) using the vectors.

\(\bar{a}\) = 3i + 4j + k, \(\bar{b}\) = 2i – 7 j – 3k and \(\bar{c}\) = 2i + 3j – 9k.

Answer:

\(\bar{a}+\bar{b}\) = 3i + 4j + k + 2i – 7j – 3k = 5i – 3j – 2k

\(\bar{a}-\bar{b}\) = 3i + 4j + k – (2i – 7j -3k) = i + 11j + 4k

\(\bar{b}+\bar{c}\) = 2i – 7j -3k + 2i +3j – 9k

= 4i – 4j – 12k.

Question 2.

- Find the vector passing through the point A( 1, 2, -3) and B(-1, -2, 1).
- Find the direction cosines along AB.

Answer:

1. \(\overline{A B}\) = \(\overline{O B}\) – \(\overline{O A}\) = -i – 2j + k – (i + 2j – 3k) = -2i – 4j + 4k.

2. Unit Vector

Direction cosines are \(\frac{-2}{6}\), \(\frac{-4}{6}\), \(\frac{4}{6}\).

Question 3.

Show that the points A, B and C with position vectors \(\bar{a}\) = 3i – 4j – 4k, \(\bar{b}\) = 2i – j + k and \(\bar{c}\) = i – 3j – 5k respectively from the vertices of a right angled triangle.

Answer:

41 = 35 + 6 ⇒ BC^{2} = AB^{2} + CA^{2}

Hence right angled triangle.

Question 4.

Prove that \([\bar{a}+\bar{b} \bar{b}+\bar{c} \bar{c}+\bar{a}]=2[\bar{a} \bar{b} \bar{c}]\).

Answer:

LHS

Note: If \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are coplanar then so is \([\bar{a}+\bar{b} \bar{b}+\bar{c} \bar{c}+\bar{a}]\).

Question 5.

Consider the vector \(\bar{p}\) = 2i – j + k. Find two vectors \(\bar{q}\) and \(\bar{r}\) such that \(\bar{p}\), \(\bar{q}\) and \(\bar{r}\) are mutually perpendicular.

Answer:

Find a vector \(\bar{q}\) such that \(\bar{p} \cdot \bar{q}\) = 0, for this use any \(\bar{q}\) whose two components are randomly selected. Let \(\bar{q}\) = 2i + 2j + xk

\(\bar{p} \cdot \bar{q}\) = (2i – j + k) . (2i + 2 j + xk) = 0

⇒ 4 – 2 + x = 0 ⇒ x = -2

= 6j + 6k.

Question 6.

Answer:

= i(-12 + 7) – j(-9 – 2) + k(-21 – 8)

= -5i + 11j – 29k

= i(63 + 9) – j(-18 + 6) + k(6 – 14)

= 72i + 12 j – 8k.

Question 7.

If \(\bar{a}\) = 3i + j + 2k,

(i) Find the magnitude of \(\bar{a}\). (1)

(ii) If the projection of \(\bar{a}\) on another vector \(\bar{b}\) is \(\sqrt{14}\), which among the following could be \(\bar{b}\) ? (1)

(a) i + j + k

(b) 6i + 2j + 4k

(c) 3i – j + 2k

(d) 2i + 3j + k

(iii) If \(\bar{a}\) makes an angle 60° with a vector \(\bar{c}\), find the projection of \(\bar{a}\) on \(\bar{c}\) (1)

Answer:

(ii) Since projection of \(\bar{a}\) on another vector \(\bar{b}\) and magnitude of \(\bar{a}\) is \(\sqrt{14}\), then \(\bar{a}\) and \(\bar{b}\) are parallel, (b) 6i + 2j + 4k.

(iii) Projection of \(\bar{a}\) on \(\bar{c}\)

= |\(\bar{a}\)|cos60° = \(\sqrt{14}\) × \(\frac{1}{2}\) = \(\frac{\sqrt{14}}{2}\).

Question 8.

(i) The projection of the vector 2i + 3j + 2k on the vector i + j + k is (1)

(ii) Find the area of a parallelogram whose adjacent sides are the vectors 2i + j + k and 6i – j (2)

Answer:

(ii) Let \(\bar{a}\) = 2i + j + k, \(\bar{b}\) = i – j

= i(0 + 1) – j(0 – 1) + k(-2 – 1 ) = i + j -3k

Area =

Question 9.

(i) The angle between the vectors i + j and j + k is (1)

(a) 60°

(b) 30°

(c) 45°

(d) 90°

Answer:

(i) (a) 60°

Question 10.

Answer:

(ii) Given, \(\bar{a}\) + \(\bar{b}\) + \(\bar{a}\) = \(\bar{0}\), squaring both sides we get

### Plus Two Maths Vector Algebra Four Mark Questions and Answers

Question 1.

Let A (2, 3), B (1, 4), C (0, -2) and D (x, y) are vertices of a parallelogram ABCD.

- Write the position vectors A, B, C and D. (2)
- Find the value of x and y. (2)

Answer:

1. Position vector of A = 2i + 3 j

Position vector of B = i + 4j

Position vector of C = 0i – 2j

Position vector of D = xi + yj.

2. Since ABCD is a parallelogram, then

(1) ⇒ -i + j = -xi – (y + 2 )j

x = 1, -2 – y = 1 ⇒ y = -3

∴ D is (1, -3).

Question 2.

Find the position vector of a point R which divides the line joining the two points P and Q whose vectors i + 2j – k and -i + j + k in the ratio 2:1

- internally and
- externally.

Answer:

\(\overline{O P}\) = i + 2j – k, \(\overline{O Q}\) = -i + j + k

Let R be the position vector of the dividing point,

1.

2.

Question 3.

(i) Choose the correct answer from the bracket. If a unit vector \(\widehat{a}\) makes angles \(\frac{\pi}{4}\) with i and \(\frac{\pi}{3}\) with j and acute angle θ with k. then θ is

(a) \(\frac{\pi}{6}\),

(b) \(\frac{\pi}{4}\),

(c) \(\frac{\pi}{3}\),

(d) \(\frac{\pi}{2}\) (1)

(ii) Find a unit vector \(\widehat{a}\) (1)

(iii) Write down a unit vector in XY plane, making an angle 60°of with the positive direction of x – axis. (2)

Answer:

(i) (c), Since I = cos\(\frac{\pi}{4}\) = \(\frac{1}{\sqrt{2}}\), m = cos\(\frac{\pi}{3}\) = 1/2;

n = cos θ

l^{2} + m^{2} + n^{2} = 1

n^{2} = 1 – (\(\frac{1}{2}\))^{2} – (\(\frac{1}{\sqrt{2}}\))^{2} = 1/4

n = \(\frac{1}{2}\), cosθ = 1/2 , θ = \(\frac{\pi}{3}\).

(ii)

Question 4.

Let the vectors \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) denoted the sides of a triangle ABC.

(i) Prove that (2)

(ii) Find the projection of the vector i + 3j + 7k on the vector 7i – j + 8k (2)

Answer:

(ii) Projection of the vector i + 3j + 7k on the vector 7i – j + 8k

Question 5.

(i) If \(\bar{a}\) and \(\bar{b}\) are any two vectors, then axb is (1)

(a) a vector on the same plane where \(\bar{a}\) and \(\bar{b}\) lie.

(b) ab cosθ, if θ is the angle between them.

(c) a vector parallel to both \(\bar{a}\) and \(\bar{b}\).

(d) a vector perpendicular to both \(\bar{a}\) and \(\bar{b}\).

(ii) Let \(\bar{a}\) = 2i + 4j – 5k, \(\bar{b}\) = i + 2j + 3k. Then find a unit vector perpendicular to both \(\bar{a}\) and \(\bar{b}\). (2)

(iii) Find a vector of magnitude 5 in the direction perpendicular to both \(\bar{a}\) and \(\bar{b}\) (1)

Answer:

(i) (d) a vector perpendicular to both \(\bar{a}\) and \(\bar{b}\).

(ii) \(\bar{a}\) = 2i + 4j-5k, \(\bar{b}\) = i + 2j+3k

= i(12 + 10) – j(6 + 5) + k(4 – 4) = 22i – 11j

Therefore unit vector perpendicular to both \(\bar{a}\) and \(\bar{b}\) is

(iii) 5 × unit vector perpendicular to both \(\bar{a}\) and \(\bar{b}\)

Question 6.

Consider a vector that is inclined at an angle 45° to x-axis and 60° to y-axis

- Find the dc’s of the vector. (2)
- Find a unit vector in the direction of the above vector. (1)
- Find a vector which is of magnitude 10 units in the direction of the above vector. (1)

Answer:

1. Let l, m, n are the direction ratios.

Given that, l = cos 45° = \(\frac{1}{\sqrt{2}}\), m = cos 60° = \(\frac{1}{2}\)

⇒ l^{2} + m^{2} + n^{2} = 1

∴ the dc’s of the vector are \(\frac{1}{\sqrt{2}}\), \(\frac{1}{2}\), \(\frac{1}{2}\)

2. A unit vector in the direction of the above vector is given by li + mj + nk ⇒ \(\frac{1}{\sqrt{2}}\)i + \(\frac{1}{2}\)j + \(\frac{1}{2}\)k.

3. A vector, which is of magnitude 10 units in the direction of the above vector is given by

Question 7.

Consider the point A(2, 1, 1) and B(4, 2, 3)

- Find the vector \(\overline{A B}\) (1)
- Find the direction cosines of \(\overline{A B}\) (2)
- Find the angle made by \(\overline{A B}\) with the positive direction of x-axis. (1)

Answer:

1. \(\overline{A B}\) = 2i + j + 2k

2. |\(\overline{A B}\)| = \(\sqrt{4+1+4}\) = 3

The direction cosines are \(\frac{2}{3}\), \(\frac{1}{3}\), \(\frac{2}{3}\).

3. cos α = \(\frac{2}{3}\) ⇒ α = cos^{-1}(\(\frac{2}{3}\)).

Question 8.

If i + j + k, 2i + 5j, 3i + 2 j – 3k, i – 6j – k respectively are the position vector of points A, B,C and D. Then

- Find \(\overline{A B}\) and \(\overline{C D}\). (1)
- Find the angle between the vectors \(\overline{A B}\) and \(\overline{C D}\). (2)
- Deduce that \(\overline{A B}\) parallel to \(\overline{C D}\). (1)

Answer:

1.

2.

3. Since the angle between \(\overline{A B}\) and \(\overline{C D}\) is π they are parallel.

Question 9.

Let ABCD be a parallelogram with sides as given in the figure.

- Find area of the parallelogram. (2)
- Find the distance between the sides AB and DC. (2)

Answer:

1. Given;

\(\overline{A B}\) = i – 3j + k and \(\overline{A D}\) = i + j + k

2. Let h be the distance between the parallelsides AB and DC. Then ; Area = Base × h _____(2)

Here, Base = |\(\overline{A B}\)|

|i – 3j + k| = \(\sqrt{1+9+1}=\sqrt{11}\)

From (1) and (2)

Question 10.

Consider \(\bar{a}\) = i + 2j – 3k, \(\bar{b}\) = 3i – j + 2k, \(\bar{c}\) = 11i + 2j

- Find \(\bar{a}\) + \(\bar{b}\) and \(\bar{a}\).\(\bar{b}\) (2)
- Find the unit vector in the direction of \(\bar{a}\) + \(\bar{b}\). (1)
- Show that \(\bar{a}\) + \(\bar{b}\) and \(\bar{a}\) – \(\bar{b}\) are orthogonal. (1)

Answer:

(ii) Unit vector in the direction of

(iii) We have,

Therefore, they are orthogonal.

Question 11.

Let A (1, -1, 4), B ( 2, 1, 2 ) and C (1, -2, -3 )

- Find \(\overline{A B}\). (1)
- Find the angle between \(\overline{A B}\) and \(\overline{A C}\).(2)
- Find the area of the parallelogram formed by \(\overline{A B}\) and \(\overline{A C}\) as adjacent sides. (1)

Answer:

1. \(\overline{A B}\) = P.v of B – P.v of A

= 2 i + j + 2 k – (i – j + 4k) = i + 2 j – 2k

2. \(\overline{A C}\) = P.v of C – P.v of A

= i – 2 j – 3 k -(i – j + 4k) = – j – 7k

Let A be the angle between \(\overline{A B}\) and \(\overline{A C}\)

3.

Area of the parallelogram

### Plus Two Maths Vector Algebra Six Mark Questions and Answers

Question 1.

Using this figure answer the following questions.

- Find \(\overline{O A}\), \(\overline{O B}\), \(\overline{O C}\) (2)
- Find \(\overline{O D}\) (2)
- Find the coordinate of the vertex D. (2)

Answer:

1. \(\overline{O A}\) = (3 – 1)i + (-1 – 2)j + (7 – 3)k = 2i – 3j + 4k

\(\overline{O B}\) = (2 – 1)i + (4 – 2)j +(2 – 3)k = i + 2j – k

\(\overline{O C}\) = (4 – 1)i + (1 – 2 )j + (5 – 3 )k = 3i – j + 2 k.

2. From the figure,

3. Let the vertex of D be (x , y , z),

Then, \(\overline{O D}\) = (x – 1)i + (y – 2)j + (z – 3)k.

But we have,

\(\overline{O D}\) = 6i – 2j + 5k = (x – 1)i + (y – 2)j +(z – 3)k

x – 1 = 6 ⇒ x = 7, y – 2 = -2 ⇒ y = 0, z – 3 = 5 ⇒ z = 8.

Question 2.

OABCDEFG is a cube with edges of length 8 units and axes as shown. L, M, N are midpoints of the edges FG, GD, GB respectively.

- Find p.v’s of F, B,D and G. (1)
- Show that the angle between the main diagonals is θ = cos
^{-1}\(\left(\frac{1}{3}\right)\). (2) - Find the p.v’s of L, M, N. (1)
- Show that \(\overline{L M}+\overline{M N}+\overline{N L}=0\). (1)

Answer:

1. \(\overline{O F}\) = 8 j + 8k, \(\overline{O B}\) = 8i + 8k, \(\overline{O D}\) = 8i + 8k, \(\overline{O G}\) = 8i + 8j + 8k.

2. Consider the main diagonals \(\overline{O G}\) and \(\overline{E B}\)

3. P.V of L = \(\overline{O L}\) = 4i + 8j + 8k

P.V of M = \(\overline{O M}\) = 8i + 8j + 4k

P.V of N = \(\overline{O N}\) = 8i + 4j + 8k

4.

Question 3.

Using the figure answer the following questions

- Evaluate \(\overline{A B}\).\(\overline{A C}\) (2)
- Find \(\overline{A D}\) . (2)
- Find the coordinates of D.

Answer:

1. \(\overline{A B}\) = p.v of B – p.v of A= -4i + 0j + 3k

\(\overline{A C}\) = p.v of C – p.v of A = 0i – 4 j + 4k

\(\overline{A B}\).\(\overline{A C}\) = -4 × 0 + 0 × -4 + 3 × 4 = 12

2.

3. Let the coordinate of D be (x, y ,z)

⇒ \(\overline{A D}\) = (x – 3)i + (y – 2)j + (z – 1)k,

Question 4.

Consider the Parallelogram ABCD

- Find \(\overline{A B}\) and \(\overline{A D}\) (1)
- Find the area of the parallelogram ABCD. (1)
- Find \(\overline{A C}\). (2)
- Find co-ordinate of C. (2)

Answer:

1. \(\overline{A B}\) = p.v of B – p. v of A

= 3i + 5j + 8k – (i + 2j + k) = 2i + 3j + 7k

\(\overline{A D}\) = p.v of D – p. v of A

= i + 3j + 2k – (i + 2j + k)= 0i + j + k.

2.

3. By triangle inequality;

4. Let the co-ordinate of C be (x, y, z)

Then, \(\overline{A C}\) = (x – 1)i + (y – 2)j + (z – 1)k = 2i + 4j + 8k

x – 1 = 2 ⇒ x = 3, y – 2 = 4 ⇒ y = 6,

z – 1 = 8 ⇒ z = 9

Co-ordinate of C is (3, 6, 9).

Question 5.

Consider the following quadrilateral ABCD in which P, Q, R, S are the midpoints of the sides.

- Find \(\overline{P Q}\) and \(\overline{S R}\) in terms of \(\overline{A C}\) (2)
- Show that PQRS is a parallelogram. (2)
- If \(\bar{a}\) is any vector, prove that (2)

Answer:

1. Using triangle law of addition, we get

2. From the above explanation we have,

and parallel. Similarly, |\(\overline{S P}\)| = |\(\overline{R Q}\)|

Therefore, PQRS is a parallelogram.

3. Let \(\bar{a}\) = a_{1} i + a_{2} j + a_{3} k