1.

If vec(a),vec(b),vec(c ) be three vector of magnitude sqrt(3), 1, 2, such thatvec(a)xx(vec(a)xx vec(c ))+3vec(b)=vec(0), if theta is theangle between , then cos theta is equal to :

Answer»

`3//4`
`1//2`
`1//4`
None of these

SOLUTION :We have
`VEC(a)xx(vec(a)xx vec(c ))+3vec(B)=vec(0)`
`rArr (vec(a).vec(c ))vec(a)-(vec(a)vec(a))vec(c )+3vec(b)=vec(0)`
`rArr (2 sqrt(3)cos THETA)vec(a)-3vec(c )+ 3vec(b)=vec(0)`
`rArr (2 cos theta)vec(a)-sqrt(3)vec(c )+ sqrt(3)vec(b)=vec(0)`
`rArr |(2 cos theta)vec(a)-sqrt(3)vec(c )|=|-sqrt(3)vec(b)|`
`rArr |2cos theta vec(a)-sqrt(3)vec(c )|^(2) = |-sqrt(3)vec(b)|^(2)`
`rArr 4cos^(2)theta|vec(a)|^(2)+3|vec(c )|^(2)-4sqrt(3)cos theta (vec(a).vec(c ))=3|vec(b)|^(2)`
`rArr 12cos^(2)theta + 12-4sqrt(3)cos theta xx sqrt(3)xx 2 cos theta = 3`
`rArr 12 cos^(2)theta + 9-24 cos^(2)theta = 0`
`rArr 12 cos^(2)theta = 9 rArr cos^(2)theta = (9)/(12)`
`rArr cos theta = (3)/(4)`


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