Vectors
Vectors
Allen Star Batch
Grade 12
Question:
If the lines $\vec{r} = \vec{a} + (\vec{b} \times \vec{c})$, and $\vec{r} = \vec{b} + s(\vec{c} \times \vec{a})$ intersect (t and s are scalars) then:
$\vec{a} \cdot \vec{c} = 0$
$\vec{a} \cdot \vec{c} = \vec{b} \cdot \vec{c}$
$\vec{b} \cdot \vec{c} = 0$
None of these
Step-by-Step Solution
Key Concept: Two lines intersect if the vector from one point to another lies in both planes defined by the direction vectors.
For intersecting lines $\vec{r} = \vec{a} + t(\vec{b} \times \vec{c})$ and $\vec{r} = \vec{b} + s(\vec{c} \times \vec{a})$, at intersection we have $\vec{d}\vec{c} = \vec{b}\vec{c}$. This yields $\vec{a}\vec{c} = \vec{b}\vec{c}$, confirming the lines meet when this condition holds.
Correct Answer: 2