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Question

The scalar product of the vector ^i+^j+^k with a unit vector along the sum of vectors 2^i+4^j5^k and λ^i+2^j+3^k is equal to one. Find the value of λ.


Solution

Let a=^i+^j+^k,b=2^i+4^j5^k and c=λ^i+2^j+3^k
Now, b+c=2^i+4^i5^k+λ^i+2^j+3^k=(2+λ)^i+6^j2^k
|b+c|=(2+λ)2+(6)2+(2)2=4+λ2+4λ+36+4=λ2+4λ+44
The unit vector along (b + c), i.e., b+c|b+c|=(2+λ)^i+6^j2^kλ2+4λ+44
1(2+λ)+1(6)+1(2)λ2+4λ+44=1(2+λ)+62λ2+4λ+44=1λ+6=λ2+4λ+44λ2+12λ+36=λ2+4λ+448λ=8λ=1
Hence, the value ofλ is 1.

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