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Question

If the vector ^i+^j^k bisects the angle between the vector c and the vector3^i+4^j,then the unit vector in the direction of c,is

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Solution

a=3^i+4^jcbisectb=^i+^j^k
So, we can write
a+c=λb^a+^c=λ^b[^a=3^i+4^j9+16=3^i+4^j15=3^i+4^j5]3i5+4j5+^c=λ(^i+^j^k3)(i)
Let the angle between a & b is θ
a.b=abcosθcosθ=a.bab=3+453=153a+c=λba.b+c.b=λ(b2)|σ|acosθ+cbcosθ=3λ1×3×153+1.3×153=3λ15+15=3λ25=3λ215=λ3^i5+4^j5+^c=215(^i+^j^k)c=215^i+215^j215^k3^i54^j5c=1115^i1015^j215^kc=115(11^i+10^j+2^k)

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