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Question

Mathematics Question on Vectors

The projection of the vector 2i^+j^−3k^2\hat{i}+\hat{j}-3\hat{k} on the vector i^−2j^−k^\hat{i}-2\hat{j}-\hat{k} is

A

−314-\frac {3}{\sqrt {14}}

B

314\frac {3}{\sqrt {14}}

C

−32-\sqrt{\frac{3}{2}}

D

32\frac {3}{\sqrt {2}}

Answer

−32-\sqrt{\frac{3}{2}}

Explanation

Solution

Let a=2i^+j^−3k^a =2 \hat{ i }+\hat{ j }-3 \hat{ k } and b=i^−2j^+k^b =\hat{ i }-2 \hat{ j }+\hat{ k } projection of aa on b=a⋅b∣b∣b =\frac{ a \cdot b }{| b |} =(2i^+j^−3k^)⋅(i^−2j^+k^)12+(−2)2+12=\frac{(2 \hat{ i }+\hat{ j }-3 \hat{ k }) \cdot(\hat{ i }-2 \hat{ j }+\hat{ k })}{\sqrt{1^{2}+(-2)^{2}+1^{2}}} =2−2−36=−36=−32=\frac{2-2-3}{\sqrt{6}}=-\frac{3}{\sqrt{6}}=-\frac{\sqrt{3}}{2}