The scalar projection of \(\vec{a} = \lambda\hat{i} + \hat{j} - 2\hat{k}\) on \(\vec{b} = 2\hat{i} - \hat{j} - \lambda\hat{k}\) is \(7/3\). What is the value of \(\lambda\)?
\(2\)
The scalar projection of \(\vec{a}\) on \(\vec{b}\) is \(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\). Here \(\vec{a}\cdot\vec{b}=2\lambda-1+2\lambda=4\lambda-1\) and \(|\vec{b}|=\sqrt{4+1+\lambda^2}=\sqrt{5+\lambda^2}\). Setting \(\dfrac{4\lambda-1}{\sqrt{5+\lambda^2}}=\dfrac{7}{3}\) and testing \(\lambda=2\) gives \(\dfrac{7}{\sqrt9}=\dfrac{7}{3}\), which satisfies the equation, so \(\lambda=2\).
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