# PRODUCT OF VECTORS INTERMEDIATE FIRST YEAR 1A CHAPTER 5 PROBLEMS WITH SOLUTIONS

**Mathematics 1A Intermediate** Product of vectors some problems with solutions are given below for examination purpose.

These are very simple to understand.

Study the textbook lesson Product of Vectors very well.

Observe the example problems and solutions given in the textbook.

**You can also see**

SSC Maths text book Solutions class 10

Inter Maths 1A Solutions textbook

Inter Maths 1B textbook solutions

Inter Maths 1IA text book solutions

Inter Maths IIB text book solutions

**Product of Vectors **

Model papers for maths SSC class 10 and Inter

## VECTOR ALGEBRA PRODUCT OF VECTORS

### Very short answer questions

**Inter first year 1A class 11 maths solutios chapter 4**

1. Find the angle between the vectors i + 2j + 3k and 3i – j + 2k.

2. If a = i + 2j – 3k and b = 3i – j + 2k then show that a + b and a – b are perpendicular to each other.

3. Let a and b be non zero, non collinear vectors. If |a + b| = |a – b| then find the angle between a and b.

**Inter first year 1A class 11 maths solutios chapter 4**

5. Let a = i + j + k and b = 2i + 3j + k,find projection vector of b on a and its magnitude.

6. If a = 2i + 2j – 3k, b = 3i – j + 2k then find the angle between the vectors 2a + b and a + 2b.

7. If a = 2i – j + k, b = i – 3j – 5 k then find |a×b|.

8. If 4i + (2p/3)j + pk is parallel to the vector i + 2j +3k, find p.

9. Find the area of the parallelogram having a = 2j – k and b = – i + k as adjacent sides.

10. Find the area of the parallelogram whose diagonals are 3i + j + 2k and i – 3j + 4k.

11. Find unit vector perpendicular to the plane determined by the vectors a = 4i + 3j – k and b = 2i – 6k – 3k.

Problem 12

Let a = 2i – j + k and b = 3i + 4j – k. if theta is the angle between a and b, then find sin theta.

Problem

Compute (i – j. J – k. K – i).

Problem

if a = i – 2j – 3k, b = 2i + j – k, c = i + 3j – 2k then compute a.(b × c).

Problem

Find the volume of the parallelopiped coterminus edges i + j + k, i – j and i + 2j – k.

Problem

For non coplanar vectors a, b and c, determine p for which the vectors a + b + c, a + pb + 2c and – a + b + c are coplanar.

Problem

Show that i × ( a × i) + j × (a ×j) + k × ( a ×k) = 2a,for any vector a.

Problem

For any three vectors a, b, c, prove that [ b + c. c + a. a + b] = 2 [a, b,. c].

Problem

For any three vectors a, b. c, prove that [b ×c. c × a. a × b] = [a. b. c]^2.

### SOLUTIONS FOR JUNIOR INTER MATHS PRODUCT OF VECTORS

### Short answer questions

Problem

Prove that angle in a semi circle is a right angle by using vector method.

Problem

If P, Q, R and S are points whose position vectors are i – k, – i + 2j, 2i – 3k and 3i – 2j – k respectively then find the components of RS on PQ.

M

Problem

Show that the points (5, – 1), (7, – 4, 7), (- 1, – 3, 4) are the vertices of a rhombus by vectors.

Problem

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

Problem

If a + b + c = 0, then prove that a ×b = b × c = c × a.

Problem

Find the unit vector perpendicular to the plane passing through the points (1, 2, 3), (2, – 1, 1) and (1, 2, – 4)

Problem

If a = 2i + 3j + 4k, b = i + j – k and c = i – j + k, then compute a × ( × c) and verify that it is perpendicular to a.

M

Problem

Let a. b and c be unit vectors such that b is not parallel to c and a × (b × c) = (1/2)b. Find the angle made by a with each of b and c.

Problem

Let a = i + j + k, b = 2i – j + 3k, c = i – j and d = 6i + 2j + 3k. Express d in terms of b × c, c × a and a × b.

Problem

For any four vectors a, b, c and d, show that i. (a × b) × (c × d) = [a. b. c] b – [b. c. d] and ii. (a × b) × (c × d) = [a. b. d] c – [a. b. c] d.

Problem

a, b, c are non zero vectors and a is perpendicular to both b and c. If |a| = 2, |b|= 3, |c|= 4 and (b, c) = 2π/3, then find |[a. b. c]|.

Problem

If |b. c. d| + |c. a. d| + |a. b. d| = [a. b. c], then show that the points with position vectors a, b, c and d. are coplanar.

Problem

Find the volume of the tetrahedron whose vertices are (1, 2, 1), (3, 2, 5), (2, – 1, 0) and (- 1, 0, 1).

Problem

Prove that rhe four points 4i + 5j + k, – (j + k), 3i + 9j + 4k and – 4i + 4j + 4k are coplanar.

Problem

If a = 2i + j – k, b = – i + 2i – 4k and c = i + j + k, then find (a × b).(b × c).

**Product of vectors long answer questions**

Problem

By the vector method, prove that the perpendicular bisectors of the sides of a triangle are concurrent.

Problem

If a. b, c be three vectors. Then show that (a × b) × c = (a × c) b – (b × c) a.

Problem

Find the equation of the plane passing through the points A = (2, 3, – 1), B = (4, 5, 2) and C = (3, 6, 5).

Problem

Find the equation of the plane passing through the points A = (3, – 2, – 1) and parallel to the vectors b = i – 2j + 4k and c = 3i + 2j – 5k.

Problem

Find the shortest distance between the skew lines r = (6i + 2j + 2k) + t (i – 2j + 2k) and r = (- 4i – k) + s ( 3i – 2j + 2k) and r = (- 4i – k) + s ( 3i – 2 j – 2k).

Problem

A = (1, – 2, – 1), B = (4, 0, – 3), C = (1, 2, – 1) and D = (2, – 4, – 5), find the distance between AB and CD.

Problem

If a = i – 2j + k, b = 2i + j + k, c = i + 2j – k, find a × ( b × c) and |(a × b) × c|.

Problem

If a = 2i + j – 3k, b = i – 2j + k, b = – i + j – 4k and d = i + j + k, then compute (a × b) × ( c × d).

Problem

If a = i – 2j + 3k, b = 2i – j + k, c = i + j + 2k then find |(a × b) × c| and |a × (b × c).

Problem

Let a = 2i + j – 2k, b = i + j, if c is a vector such that a.c = |c|, |c – a| = 2√2 and the angle between a × b and c is 30°, then find the value of |a × b|× |c|.

Note : Observe the solutions and try them in your own methods.

**Some more solutions**

**You can see the solutions for junior inter maths 1A for examination purpose**

1. Functions

3. Matrices

5. Trigonometric ratios upto transformations 1

6. Trigonometric ratios upto transformations 2

8. Inverse trigonometric functions

Exercise 6(a) inter maths 1A solutions

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