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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra.

BoardSCERT, Kerala
Text BookNCERT Based
ClassPlus Two
SubjectMaths Chapter Wise Questions
ChapterChapter 10
Chapter NameVector Algebra
Number of Questions Solved48
CategoryKerala Plus Two 

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 with 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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q2

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:

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q3

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q4

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q5

= 6j + 6k.

Question 6.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q6

Answer:

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q6.1

= i(-12 + 7) – j(-9 – 2) + k(-21 – 8)

= -5i + 11j – 29k

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q6.2

= 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:

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q7

(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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q8

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

Answer:

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q8.1

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q8.2

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

Area =

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q8.3

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q9

Answer:

(i) (a) 60°

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q9.1

Question 10.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q10

Answer:

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q10.1

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 3M Q10.2

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q1

(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

  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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q2

2.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q2.1

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q3

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q3.1

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q4

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

Answer:

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q4.1

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q4.2

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q5

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q5.1

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q5.2

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q5.3

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q6

∴ 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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q6.1

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q8

2.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q8.1

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.

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q9

Answer:

1. Given;

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

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q9.1

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q9.2

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q10

(ii) Unit vector in the direction of

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q10.1

(iii) We have,

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q10.2

Therefore, they are orthogonal.

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q11

3.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q11.1

Area of the parallelogram

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 4M Q11.2

Plus Two Maths Vector Algebra Six Mark Questions and Answers

Question 1.

Using this figure answer the following questions.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q1

  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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q1.1

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.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q2

  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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q2.1

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q2.2

Question 3.

Using the figure answer the following questions

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q3

  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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q3.1

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q3.2

Question 4.

Consider the Parallelogram ABCD

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q4

  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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q4.1

3. By triangle inequality;

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q4.2

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.

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q5

  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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q5.1

Answer:

1. Using triangle law of addition, we get

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q5.2

2. From the above explanation we have,

Plus Two Maths Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q5.3

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 Chapter Wise Questions and Answers Chapter 10 Vector Algebra 6M Q5.4

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