Let a = 2hat{i} - hat{j} + 2hat{k} and b = 3h | vedclass

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(D) Given vectors are $a = 2\hat{i} - \hat{j} + 2\hat{k}$ and $b = 3\hat{i} - 2\hat{j} - 5\hat{k}$.We need to find the projection of vector $b$ on a v

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$\frac{31}{9}\hat{i} - \frac{20}{9}\hat{j} - \frac{41}{9}\hat{k}$

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(D) Given vectors are $a = 2\hat{i} - \hat{j} + 2\hat{k}$ and $b = 3\hat{i} - 2\hat{j} - 5\hat{k}$.We need to find the projection of vector $b$ on a vector perpendicular to $a$. This is equivalent to finding the component of $b$ perpendicular to $a$,which is given by $b_{\perp a} = b - 	ext{proj}_a b$.The projection of $b$ on $a$ is $	ext{proj}_a b = \left(\frac{a \cdot b}{|a|^2}ight)a$.First,calculate $a \cdot b = (2)(3) + (-1)(-2) + (2)(-5) = 6 + 2 - 10 = -2$.Next,calculate $|a|^2 = 2^2 + (-1)^2 + 2^2 = 4 + 1 + 4 = 9$.Thus,$	ext{proj}_a b = \left(\frac{-2}{9}ight)(2\hat{i} - \hat{j} + 2\hat{k}) = -\frac{4}{9}\hat{i} + \frac{2}{9}\hat{j} - \frac{4}{9}\hat{k}$.Now,$b_{\perp a} = (3\hat{i} - 2\hat{j} - 5\hat{k}) - (-\frac{4}{9}\hat{i} + \frac{2}{9}\hat{j} - \frac{4}{9}\hat{k})$.$b_{\perp a} = (3 + \frac{4}{9})\hat{i} + (-2 - \frac{2}{9})\hat{j} + (-5 + \frac{4}{9})\hat{k}$.$b_{\perp a} = \frac{31}{9}\hat{i} - \frac{20}{9}\hat{j} - \frac{41}{9}\hat{k}$.

Let $a = 2\hat{i} - \hat{j} + 2\hat{k}$ and $b = 3\hat{i} - 2\hat{j} - 5\hat{k}$ be two vectors. Then the projection vector of $b$ on a vector perpendicular to $a$ is

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