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Let a , b and c be three unit vector...

Let a , b and c be three unit vectors out of which vectors b and c are non -parallel. If `alpha " and " beta` are the angles which vector a makes with vectors b and c respectively and `axx (b xx c) = (1)/(2) b,` Then `|alpha -beta|` is equal to

A

`30^(@)`

B

`45^(@)`

C

`90^(@)`

D

`60^(@)`

Text Solution

Verified by Experts

The correct Answer is:
A
Given `a xx (b xx c) =(1)/(2) b rArr (a . c) -(a .b) c= (1)/() b`
`[:' a xx (bxxc) = (a.b) c =(1)/(2) b`
On comparing both sides we get
`a.c =(1)/(2) ......(i)`
`" and " a.b =0 .....(i)`
`:' a,b ` and c are unit vectors and angle between a and b is alpha and angle between a and c is `beta ` so
`|a||c| cos beta = (1)/(2)`
`rArr cos beta =(1)/(2) [ :' |a| =1= |c|]`
`rArr beta = (pi)/(3) .....(iii) [:' cos .(pi)/(3) = (1)/(2)]`
and `|a||b| cos alpha =0` [from Eq. (ii)]
`rArr alpha= (pi)/(2) .....(iv)`
From Eq (iii) and (iv) we get
`|alpha - beta|= (pi)/(2) -(pi)/(3) =(pi)/(6) =30^(@)`
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