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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`
Solution
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)`
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