Question

Find a unit vector perpendicular to each of the vectors `veca + vecb  "and"  veca - vecb  "where"  veca = 3hati + 2hatj + 2hatk and vecb = i + 2hatj - 2hatk` 

Answer 1

Let the unit vector be λ

λ = `λ_1hati + λ_2hatj + λ_3hatk`

Now, `veca + vecb =  4hati + 4hatj + 0hatk`

`veca - vecb =  2hati + 0hatj + 4hatk`

Now, `(λ_1hati + λ_2hatj + λ_3hatk) . ( 4hati + 4hatj + 0hatk) = 0`

⇒  4λ₁ + 4λ₂ = 0

⇒ λ₁ = λ₂                              ...(i)

`(λ_1hati + λ_2hatj + λ_3hatk) . (2hati + 0hatj + 4hatk)= 0`

⇒ ( 2λ₁ + 4λ₃ ) = 0

⇒ λ₁ = - 2λ₃                     ...(ii)

Now, λ₁ = λ₂  and λ₁ = - 2λ₃
λ₂ = - λ₁

`λ_3 = - (1)/(2) λ_1`

Let  λ₁ = c (say)

λ₂ = - c

λ₃ = `- (1)/(2)` c

`λ = chati - chatj - (1)/(2)chatk`

 `hatλ = λ/|λ| = ( chati - chatj - (1)/(2) chatk)/sqrt(c^2 + (-c)^2 + (1/2 c)^2) = (c( hati - hatj - (1)/(2) hatk))/((3c)/2)`

`hatλ = (2)/(3) ( hati - hatj - (1)/(2) hatk)`