Let a= i- j+2 k, and b satisfy a b=2 i- k, a b=3.

Question

Let $\vec a=\hat i-\hat j+2\hat k$, and $\vec b$ satisfy $\vec a	imes\vec b=2\hat i-\hat k$, $\vec a\cdot\vec b=3$. The projection of $\vec b$ on $\vec a-\vec b$ is:

Accepted Answer

$$|\vec a-\vec b|^2 = |\vec a|^2+|\vec b|^2-2\vec a\cdot\vec b = 6+\frac73-6=\frac73$$

$$\text{Projection of }\vec b\text{ on }\vec a-\vec b = \frac{\vec b\cdot(\vec a-\vec b)}{|\vec a-\vec b|} = \frac{\vec a\cdot\vec b-|\vec b|^2}{|\vec a-\vec b|} = \frac{3-7/3}{\sqrt{7/3}} = \frac{2/3}{\sqrt{7/3}} = \frac2{\sqrt{21}}$$

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