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

Question

Let $\vec a=2\hat i-\hat j+5\hat k$, $\vec b=\alpha\hat i+\beta\hat j+2\hat k$. If $\big[(\vec a	imes\vec b)	imes\hat i\big]\cdot\hat k = \dfrac{23}2$, then $|\vec b	imes2\hat j|$ is:

Accepted Answer

$(\vec a\times\vec b)\times\hat i = (\vec a\cdot\hat i)\vec b - (\vec b\cdot\hat i)\vec a = 2\vec b-\alpha\vec a$.

Dot with $\hat k$: $2(2)-\alpha(5) = \dfrac{23}2 \implies 4-5\alpha=\dfrac{23}2 \implies \alpha=-\dfrac32$.

$$\vec b\times2\hat j = 2(\alpha,\beta,2)\times(0,1,0) = 2(-2,0,\alpha) = (-4,0,2\alpha)$$
$$|\vec b\times2\hat j| = \sqrt{16+4\alpha^2} = \sqrt{16+9} = 5$$

Solutions · 1