Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade None
Question:
If $\vec{r} = l(\vec{b} \times \vec{c}) + m(\vec{c} \times \vec{a}) + n(\vec{a} \times \vec{b})$ and $[\vec{b}\vec{c}\vec{c}] = 2$, then $l + m + n$ is equal to:
\vec{r} \cdot (\vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{a} \times \vec{b})
1/2 \vec{r} \cdot (\vec{a} + \vec{b} + \vec{c})
(\vec{a}\vec{b}\vec{c})
None of these
Step-by-Step Solution
Key Concept: Scalar triple product allows decomposition of any vector as linear combination of basis vectors.
Taking dot products of vector $\vec{r}$ with $\vec{a}$, $\vec{b}$, and $\vec{c}$ respectively: $l = \frac{\vec{r} \cdot \vec{a}}{[\vec{a}\vec{b}\vec{c}]} = \frac{2}{[\vec{a}\vec{b}\vec{c}]}$, $m = \frac{\vec{r} \cdot \vec{b}}{[\vec{a}\vec{b}\vec{c}]} = \frac{2}{[\vec{a}\vec{b}\vec{c}]}$, $n = \frac{\vec{r} \cdot \vec{c}}{[\vec{a}\vec{b}\vec{c}]} = \frac{2}{[\vec{a}\vec{b}\vec{c}]}$.
Correct Answer: 2