Solved Examples on Vectors

Illustration 1: Let A, B and C be vectors of length 3, 4, 5 respectively. Let A be perpendicular to B + C, B to C + A and C to A + B. then find the length of the vector A + B + C.

Solution: Given that |A| = 3, |B| = 4 and |C| = 5

Since, A.(B + C) = B.(C + A) = C. (A + B) ….. (1)

Hence, | A + B + C |² = |A|² + |B|² + |C|² + 2 (A.B + B.C + C.A)

= 9 + 16 + 25 + 0

A.B + B.C + C.A = 0….. (follows from eq (1))

Therefore, | A + B + C |² = 50

Hence, | A + B + C | = 5√2

Illustration 2: If the vector a = i+j+k, the vector b = 4i + 3j + 4k and c = i + αj + βk are linearly dependent vectors and |c| = √3 then,

1. α = 1, β = -1                                            2. α = 1, β = ±1

3. α = -1, β = ±1                                            α = ±1, β = 1 

Solution: Since a, b and c are given to be linearly dependent vectors, so this also implies that the vectors are coplanar.

So, a. (b x c) = 0

Solving this with the help of determinants, and applying the column operation

C₃ – C₁, we get

(β-1) (3-4) = 0.

This gives β = 1.

Also, |c| = √3

√ 1+ α² + β² = √3

√ 1+ α² +1 = √3

√α² + 2 = √3

α² + 2 = 3 so α = ±1.

Illustration 3: If a = (i+j+k), a.b = 1 and a x b = j-k, then b is (2003)

1. (i-j+k)                                                          2. 2j-k) 

3. i                                                                    4. 2i

Solution: We know that a x (a x b) = (a.b) a – (a.a) b

Hence, (i+j+k) x (j-k) = (i + j + k) – (√3)² b

This gives -2i + j + k = i + j + k – 3b

Hence, 3b = 3i

And this yields b = i.

*Note : Vectors have been denoted by bold letters.