JEE MAIN - Mathematics (2023 - 1st February Evening Shift - No. 4)
Let $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$$ be two vectors. Then which one of the following statements is TRUE ?
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction
of $$\vec{b}$$.
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction
of $$\vec{b}$$.
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
Explanation
$\begin{aligned} & \text { Projection of }\vec{a} \text { on } \vec{b} =\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \\\\ & = \frac{(5 \hat{i}-\hat{j}-3 \hat{k}) \cdot(\hat{i}+3 \hat{j}+5 \hat{k})}{\sqrt{1^2+3^2+5^2}}=\frac{5-3-15}{\sqrt{35}} \\\\ & = \frac{-13}{\sqrt{35}}\end{aligned}$
Negative sign indicates that direction of the projection vector is opposite to the direction of $$\vec{b}$$.
Negative sign indicates that direction of the projection vector is opposite to the direction of $$\vec{b}$$.
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