Vector Algebra
Coplanar Non-Coplanar Conditions
Grade 12
Question:
<p>Let \(\hat{p},\hat{q},\hat{r}\) be three mutually orthogonal unit vectors.
If vector \(\vec{s}\) satisfies
\[\hat{p}\times(\vec{s}\times\hat{p})+\hat{q}\times(\vec{s}\times\hat{q})+\hat{r}\times(\vec{s}\times\hat{r})=\lambda\vec{s},\]
find \(\lambda\).</p>
Step-by-Step Solution
Key Concept: Use the identity: u \times (v \times u) = |u|^2v - (u \cdot v)u (BAC–CAB). Sum over the three orthonormal vectors.
By BAC-CAB: $\hat{p}\times(\vec{s}\times\hat{p})=|\hat{p}|^2\vec{s}-(\hat{p}\cdot\vec{s})\hat{p}=\vec{s}-(\hat{p}\cdot\vec{s})\hat{p}$.
Summing over $\hat{p},\hat{q},\hat{r}$:
$\sum[\vec{s}-(\hat{e}\cdot\vec{s})\hat{e}]=3\vec{s}-\sum(\hat{e}\cdot\vec{s})\hat{e}=3\vec{s}-\vec{s}=2\vec{s}$.
(The last step uses that $\{p,q,r\}$ is an orthonormal basis, so
$\sum_e(\hat{e}\cdot\vec{s})\hat{e}=\vec{s}$ -- this is just resolving s into components.)
Therefore $\lambda=\boxed{2}$.
Correct Answer: 2