Find the vector projection of on and the scalar component of on .
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
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Given $\vec{A} = \hat{i} - 2\hat{j} - 2\hat{k}$ and $\vec{B} = 6\hat{i} + 3\hat{j} + 2\hat{k}$.
$|\vec{A}| = \sqrt{1^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$.
$\vec{B} \cdot \vec{A} = (6)(1) + (3)(-2) + (2)(-2) = 6 - 6 - 4 = -4$.
1. Scalar component of $\vec{B}$ on $\vec{A}$:
$\frac{\vec{B} \cdot \vec{A}}{|\vec{A}|} = \frac{-4}{3} = -\frac{4}{3}$.
2. Vector projection of $\vec{B}$ on $\vec{A}$:
$\left(\frac{\vec{B} \cdot \vec{A}}{|\vec{A}|^2}\right)\vec{A} = \frac{-4}{9}(\hat{i} - 2\hat{j} - 2\hat{k}) = -\frac{4}{9}\hat{i} + \frac{8}{9}\hat{j} + \frac{8}{9}\hat{k}$.
From ISC 2027 Specimen Mathematics Paper 1, question 7(i).