Vectors & 3D Geometry
Vector equation system; magnitude
MJMT_Full_Test_10
Grade 12
Question:
Let $\vec{a}=3\hat i-4\hat j+5\hat k$, $\vec{b}=\hat i+2\hat j-2\hat k$, $\vec{c}=5\hat i-12\hat j-13\hat k$. If $\vec{r}\times\vec{b}=\vec{b}\times(\vec{a}-\vec{c})$ and $\vec{r}\cdot(\vec{a}+\vec{c})=0$, then $|\vec{r}-36\vec{b}|^2$ is
Step-by-Step Solution
Key Concept: $\vec{r}\times\vec{b}=\vec{b}\times(\vec{a}-\vec{c})=(\vec{a}-\vec{c})\times\vec{b}\cdot(-1)$... rearrange to $(\vec{r}+\vec{a}-\vec{c})\times\vec{b}=\vec{0}$ so $\vec{r}=\vec{c}-\vec{a}+\lambda\vec{b}$.
$|\vec{r}-36\vec{b}|^2=392$.
Correct Answer: 392