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Solve the following.
Consider the two vectors and . Find the projection of on if .
Consider the two vectors $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\vec{b} = 5\hat{i} + q\hat{j} - \hat{k}$.
Find the projection of $\vec{a}$ on $\vec{b}$ if $q = 1$.
Answer
Answer
AIFor $q = 1$, $\vec{b} = 5\hat{i} + \hat{j} - \hat{k}$: $\vec{a}\cdot\vec{b} = 10 - 3 - 4 = 3$ and $|\vec{b}| = \sqrt{25 + 1 + 1} = 3\sqrt{3}$.
Projection of $\vec{a}$ on $\vec{b} = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \dfrac{3}{3\sqrt{3}} = \dfrac{1}{\sqrt{3}}$.
Final answer: $\dfrac{1}{\sqrt{3}}$
From ISC 2025 Improvement Mathematics Paper 1, question 17(iii).
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