If $\vec{f}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{g}=2 \hat{i}-\hat{j}+3 \hat{k}$ then the projection vector of…

If $\vec{f}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{g}=2 \hat{i}-\hat{j}+3 \hat{k}$ then the projection vector of $\vec{f}$ on $\vec{g}$ is
  1. $\frac{2}{7}(\hat{i}+\hat{j}+\hat{k})$
  2. $\frac{2}{7}(2 \hat{i}-\hat{j}+3 \hat{k})$
  3. $\frac{1}{3}(\hat{i}+\hat{j}+\hat{k})$
  4. $\frac{1}{14}(2 \hat{i}-\hat{j}+3 \hat{k})$

Solution

$\vec{f}=\hat{i}+\hat{j}+\hat{k} \text { and } \vec{g}=2 \hat{i}-\hat{j}+3 \hat{k}$
Projection of $\vec{f}$ on $\vec{g}=\frac{\vec{f} \cdot \vec{g}}{|\vec{g}|^2} \vec{g}$ $=\frac{(2-1+3)}{(4+1+9)}(2 \hat{i}-\hat{j}+3 \hat{k})=\frac{2}{7}(2 \hat{i}-\hat{j}+3 \hat{k})$

Asked in: AP EAMCET 2024 (22 May Shift 1)

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