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Solve the following.
Find the vector projection of B⃗ = 6î + 3ĵ + 2k̂ on A⃗ = î - 2ĵ - 2k̂ and the scalar component…
Find the vector projection of $\vec{B} = 6\hat{i} + 3\hat{j} + 2\hat{k}$ on $\vec{A} = \hat{i} - 2\hat{j} - 2\hat{k}$ and the scalar component of $\vec{B}$ on $\vec{A}$.
Answer
Answer
Official answer key$\vec{B}\cdot\vec{A} = 6 - 6 - 4 = -4$ and $|\vec{A}|^2 = 1 + 4 + 4 = 9$, so $|\vec{A}| = 3$.
Vector projection of $\vec{B}$ on $\vec{A}$ $= \frac{\vec{B}\cdot\vec{A}}{|\vec{A}|^2}\vec{A} = -\frac{4}{9}\left(\hat{i} - 2\hat{j} - 2\hat{k}\right)$.
Scalar component of $\vec{B}$ on $\vec{A}$ $= \frac{\vec{B}\cdot\vec{A}}{|\vec{A}|} = -\frac{4}{3}$.