If the projection of the vector \(\lambda\hat{\imath}+2\hat{\jmath}-\hat{k}\) on the vector \(2\hat{\imath}-\hat{\jmath}+2\hat{k}\) is 2 units, then the value of \(\lambda\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 5
Let \(\vec{a}=\lambda\hat{\imath}+2\hat{\jmath}-\hat{k}\) and \(\vec{b}=2\hat{\imath}-\hat{\jmath}+2\hat{k}\). The projection of \(\vec{a}\) on \(\vec{b}\) is \(\displaystyle\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\).
\(\vec{a}\cdot\vec{b} = 2\lambda-2-2 = 2\lambda-4\)
\(|\vec{b}| = \sqrt{4+1+4} = 3\)
Setting the projection equal to 2:
\( \displaystyle \frac{2\lambda-4}{3} = 2 \)
\(2\lambda-4 = 6\)
\(\lambda = 5\)