Question

# Let →a and →c be unit vectors and |→b|=4 with →a×→b=2→a×→c. The angle between →a and →c is cos−1(14). If →b−2→c=λ→a, then λ is equal to13,232,33,−413,12

Solution

## The correct option is C 3,−4It is given that the angle between →a and →c is cos−1(14). So, →a⋅→c=|→a||→c|cos(cos−1(14)) ⇒→a⋅→c=14     ⋯(1) →b−2→c=λ→a Taking dot products with →a, we have →a⋅→b−2(→a⋅→c)=λ(→a⋅→a) ⇒→a⋅→b−12=λ ⇒→a⋅→b=12+λ Similarly, taking dot products with →b and →c →b⋅→c=8−λ22−λ4     ⋯(2)  and →b⋅→c−2=λ(→a⋅→c)     ⋯(3) From equations (1),(2) and (3), we get 8−λ22−λ4−2=λ(14) ⇒λ=3,−4

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