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

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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.

  1. Find the vector passing through the point A( 1, 2, -3) and B(-1, -2, 1).
  2. 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
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 1
Direction cosines are \(\frac{-2}{6}\), \(\frac{-4}{6}\), \(\frac{4}{6}\).

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

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:
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 2
41 = 35 + 6 ⇒ BC2 = AB2 + CA2
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
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 3
Note: If \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) are coplanar then so is \([\bar{a}+\bar{b} \bar{b}+\bar{c} \bar{c}+\bar{a}]\).

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

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
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 4
= 6j + 6k.

Question 6.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 5
Answer:
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 6
= i(-12 + 7) – j(-9 – 2) + k(-21 – 8)
= -5i + 11j – 29k
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 7
= i(63 + 9) – j(-18 + 6) + k(6 – 14)
= 72i + 12 j – 8k.

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

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:
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 8

(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)
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 9
(ii) Find the area of a parallelogram whose adjacent sides are the vectors 2i + j + k and 6i – j (2)
Answer:
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 10

(ii) Let \(\bar{a}\) = 2i + j + k, \(\bar{b}\) = i – j
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 11
= i(0 + 1) – j(0 – 1) + k(-2 – 1 ) = i + j -3k
Area =
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 12

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

Question 9.
(i) The angle between the vectors i + j and j + k is (1)
(a) 60°
(b) 30°
(c) 45°
(d) 90°
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 13
Answer:
(i) (a) 60°

Plus Two Maths Vector Algebra 3 Mark Questions and Answers 14

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

Question 10.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 15
Answer:
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 16

(ii) Given, \(\bar{a}\) + \(\bar{b}\) + \(\bar{a}\) = \(\bar{0}\), squaring both sides we get
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 17

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.

  1. Write the position vectors A, B, C and D. (2)
  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
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 18
(1) ⇒ -i + j = -xi – (y + 2 )j
x = 1, -2 – y = 1 ⇒ y = -3
∴ D is (1, -3).

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

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

  1. internally and
  2. 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.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 19

2.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 20

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

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
l2 + m2 + n2 = 1
n2 = 1 – (\(\frac{1}{2}\))2 – (\(\frac{1}{\sqrt{2}}\))2 = 1/4
n = \(\frac{1}{2}\), cosθ = 1/2 , θ = \(\frac{\pi}{3}\).

(ii)
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 21
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 22

Question 4.
Let the vectors \(\bar{a}\), \(\bar{b}\), \(\bar{c}\) denoted the sides of a triangle ABC.
(i) Prove that (2)
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 23
(ii) Find the projection of the vector i + 3j + 7k on the vector 7i – j + 8k (2)
Answer:
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 24

(ii) Projection of the vector i + 3j + 7k on the vector 7i – j + 8k
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 25

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

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
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 26
= i(12 + 10) – j(6 + 5) + k(4 – 4) = 22i – 11j
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 27
Therefore unit vector perpendicular to both \(\bar{a}\) and \(\bar{b}\) is
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 28

(iii) 5 × unit vector perpendicular to both \(\bar{a}\) and \(\bar{b}\)
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 29

Question 6.
Consider a vector that is inclined at an angle 45° to x-axis and 60° to y-axis

  1. Find the dc’s of the vector. (2)
  2. Find a unit vector in the direction of the above vector. (1)
  3. 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° = \(\frac{1}{2}\)
⇒ l2 + m2 + n2 = 1
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 30
∴ 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
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 31

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

Question 7.
Consider the point A(2, 1, 1) and B(4, 2, 3)

  1. Find the vector \(\overline{A B}\) (1)
  2. Find the direction cosines of \(\overline{A B}\) (2)
  3. 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

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

Answer:
1.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 32

2.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 33

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

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

Question 9.
Let ABCD be a parallelogram with sides as given in the figure.

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

Plus Two Maths Vector Algebra 3 Mark Questions and Answers 34
Answer:
1. Given;
\(\overline{A B}\) = i – 3j + k and \(\overline{A D}\) = i + j + k
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 35

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)
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 36

Question 10.
Consider \(\bar{a}\) = i + 2j – 3k, \(\bar{b}\) = 3i – j + 2k, \(\bar{c}\) = 11i + 2j

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

Answer:
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 37

(ii) Unit vector in the direction of
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 38

(iii) We have,
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 39
Therefore, they are orthogonal.

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

Question 11.
Let A (1, -1, 4), B ( 2, 1, 2 ) and C (1, -2, -3 )

  1. Find \(\overline{A B}\). (1)
  2. Find the angle between \(\overline{A B}\) and \(\overline{A C}\).(2)
  3. 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 \(\overline{A C}\)
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 40

3.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 41
Area of the parallelogram
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 42

Plus Two Maths Vector Algebra Six Mark Questions and Answers

Question 1.
Using this figure answer the following questions.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 43

  1. Find \(\overline{O A}\), \(\overline{O B}\), \(\overline{O C}\) (2)
  2. Find \(\overline{O D}\) (2)
  3. 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,
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 44

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.

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

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.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 45

  1. Find p.v’s of F, B,D and G. (1)
  2. Show that the angle between the main diagonals is θ = cos-1\(\left(\frac{1}{3}\right)\). (2)
  3. Find the p.v’s of L, M, N. (1)
  4. 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 \(\overline{E B}\)
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 46

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.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 47

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

Question 3.
Using the figure answer the following questions
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 48

  1. Evaluate \(\overline{A B}\).\(\overline{A C}\) (2)
  2. Find \(\overline{A D}\) . (2)
  3. 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.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 49

3. Let the coordinate of D be (x, y ,z)
⇒ \(\overline{A D}\) = (x – 3)i + (y – 2)j + (z – 1)k,
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 50

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

Question 4.
Consider the Parallelogram ABCD
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 51

  1. Find \(\overline{A B}\) and \(\overline{A D}\) (1)
  2. Find the area of the parallelogram ABCD. (1)
  3. Find \(\overline{A C}\). (2)
  4. 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.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 52

3. By triangle inequality;
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 53

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).

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

Question 5.
Consider the following quadrilateral ABCD in which P, Q, R, S are the midpoints of the sides.
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 54

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

Plus Two Maths Vector Algebra 3 Mark Questions and Answers 55
Answer:
1. Using triangle law of addition, we get
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 56

2. From the above explanation we have,
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 57
and parallel. Similarly, |\(\overline{S P}\)| = |\(\overline{R Q}\)|
Therefore, PQRS is a parallelogram.

3. Let \(\bar{a}\) = a1 i + a2 j + a3 k
Plus Two Maths Vector Algebra 3 Mark Questions and Answers 58

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