Question

Find the projection of vector \(\vec{b}=2\hat{i}+2\sqrt{2} \,\hat{j}-2\hat{k}\) on the vector \(\vec{a}=\hat{i}-\hat{j}-\sqrt{2} \,\hat{k}\).

(a) 2

(b) \(\sqrt{2}\)

(c) 1

(d) \(2\sqrt{2}\)

I had been asked this question in an interview.

The doubt is from Product of Two Vectors-1 topic in division Vector Algebra of Mathematics – Class 12

Answer 1

Right answer is (b) \(\sqrt{2}\)

For explanation: The projection of vector \(\vec{b}\) on the vector \(\vec{b}\) is given by \(\frac{1}{|\vec{a}|} (\vec{a}.\vec{b})\)

\(|\vec{a}|=\sqrt{(1)^2+(-1)^2+(-\sqrt{2})^2}=\sqrt{1+1+2}=\sqrt{4}\)=2

Also, \(\vec{a}.\vec{b}=2(1)+2\sqrt{2} \,(-1)-2(-\sqrt{2})=2-2\sqrt{2}+2\sqrt{2}\)=2

Therefore, the projection of vector \(\hat{i}-\hat{j}-\sqrt{2} \,\hat{k}\) on the vector \(\vec{b}=2\hat{i}+2\sqrt{2}\hat{j}-2\hat{k}\)  is

\(\frac{1}{|\vec{a}|} (\vec{a}.\vec{b})=\frac{1}{2}\) (2)=1.