The vectors a and b are not perpendicular, and c and d are two vectors satisfying b c= b d…

Question

The vectors $\vec a$ and $\vec b$ are not perpendicular, and $\vec c$ and $\vec d$ are two vectors satisfying $\vec b	imes\vec c=\vec b	imes\vec d$ and $\vec a\cdot\vec d=0$. Then the vector $\vec d$ is equal to:

Accepted Answer

Cross both sides with $\vec a$: $\vec a\times(\vec b\times\vec c)=\vec a\times(\vec b\times\vec d)$

$$(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c = (\vec a\cdot\vec d)\vec b-(\vec a\cdot\vec b)\vec d = -(\vec a\cdot\vec b)\vec d\quad(\text{since }\vec a\cdot\vec d=0)$$

$$\vec d = \vec c - \frac{\vec a\cdot\vec c}{\vec a\cdot\vec b}\vec b$$

Solutions · 1