Let a=2 i-7 j+5 k, b= i+ k, c= i+2 j-3 k.

Question

Let $\vec a=2\hat i-7\hat j+5\hat k$, $\vec b=\hat i+\hat k$, $\vec c=\hat i+2\hat j-3\hat k$. If $\vec r	imes\vec a=\vec c	imes\vec a$ and $\vec r\cdot\vec b=0$, then $\vec r$ is equal to (magnitude):

Accepted Answer

$\vec r=\vec c+\lambda\vec a$. $\vec r\cdot\vec b=0$: $\vec c\cdot\vec b+\lambda(\vec a\cdot\vec b)=0$.

$\vec c\cdot\vec b=1+0-3=-2$, $\vec a\cdot\vec b=2+0+5=7 \Rightarrow \lambda=\dfrac27$.

$$\vec r = \left(1+\frac47,\ 2-\frac{14}7,\ -3+\frac{10}7\right) = \left(\frac{11}7,0,-\frac{11}7\right)$$
$$|\vec r| = \sqrt{\frac{121+121}{49}} = \frac{11\sqrt2}7$$

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