Vectors
Vectors
Allen Star Batch
Grade 12
Question:
If the non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendiculars to each other then the solution of the equation $\vec{r} \times \vec{a} = \vec{b}$ is given by:
$\vec{r} = x\vec{a} + \frac{1}{a\cdot a}(\vec{a} \times \vec{b}); x \in \mathbb{R}$
$\vec{r} = x\vec{b} + \frac{1}{b\cdot b}(\vec{a} \times \vec{b}); x \in \mathbb{R}$
$\vec{r} = x\vec{a} \times \vec{b}; x \in \mathbb{R}$
None of these
Step-by-Step Solution
Key Concept: Vector triple product combined with orthogonality condition isolates the unknown vector.
Given $\vec{a} \perp \vec{b}$ so $\vec{a} \cdot \vec{b} = 0$. Taking cross product of $\vec{r} \times \vec{a} = \vec{b}$ with $\vec{a}$: $\vec{a} \times (\vec{r} \times \vec{a}) = \vec{a} \times \vec{b}$. Using vector triple product formula and simplifying with the perpendicularity condition yields $\vec{r} = \frac{(\vec{a} \cdot \vec{r})\vec{a} + (\vec{a} \times \vec{b})}{\vec{a} \cdot \vec{a}}$.
Correct Answer: 1