Vectors
Vectors
DAILY_CHALLENGE
Grade 12
Question:
Given that $\vec{a}$ is perpendicular to $\vec{b}$ and p is a non-zero scalar, if $p\vec{r} + (\vec{r} \cdot \vec{b})\vec{a} = \vec{c}$ then $\vec{r} = $:
$\vec{r} = \frac{\vec{c}}{p} - \frac{c\vec{b}}{p^2}\vec{a}$
$\vec{r} = \frac{\vec{c}}{p} - \frac{c\vec{b}}{p^2}\vec{a}$
$\vec{r} = \frac{\vec{c}}{p} - \frac{c\vec{b}}{p}\vec{a}$
None of these
Step-by-Step Solution
Key Concept: Systematic dot products with known vectors isolate scalar coefficients and solve for the unknown vector.
Given $p\vec{r} + (\vec{r} \cdot \vec{b})\vec{a} = \vec{c}$, taking dot product with $\vec{b}$ gives $p\vec{r} \cdot \vec{b} + (\vec{r} \cdot \vec{b})\vec{a} \cdot \vec{b} = \vec{c} \cdot \vec{b}$. Solving yields $\vec{r} = \vec{b} + \frac{\vec{c} \cdot \vec{b}}{p}\vec{a}$.
Correct Answer: 3