Let a= j- k and c= i- j- k.

Question

Let $\vec a=\hat j-\hat k$ and $\vec c=\hat i-\hat j-\hat k$. Then the vector $\vec b$ satisfying $\vec a	imes\vec b+\vec c=\vec 0$ and $\vec a\cdot\vec b=3$ is:

Accepted Answer

Let $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$. $\vec a\times\vec b = -\vec c = -\hat i+\hat j+\hat k$:

$$\vec a\times\vec b = (\hat j-\hat k)\times(b_1\hat i+b_2\hat j+b_3\hat k) = (b_2+b_3)\hat i-b_1\hat j-b_1\hat k$$

Comparing: $b_2+b_3=-1$, $b_1=-1$. Also $\vec a\cdot\vec b=b_2-b_3=3$. Solving: $b_2=1$, $b_3=-2$.

$$\vec b = -\hat i+\hat j-2\hat k$$

Solutions · 1