3D Geometry
Direction of line joining points on two skew lines
MJMT_Full_Test_10
Grade 12
Question:
If point $A$ lies on $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and point $B$ lies on $\dfrac{x-2}{3}=\dfrac{y-4}{7}=\dfrac{z-6}{6}$, then $\overrightarrow{AB}$ cannot be parallel to
$\hat{i}+\hat{j}+2\hat{k}$
$\hat{i}+\hat{j}+\hat{k}$
$\hat{i}+\hat{j}+3\hat{k}$
$\hat{i}+\hat{j}+4\hat{k}$
Step-by-Step Solution
Key Concept: $\overrightarrow{AB}$ lies in the plane spanned by $\vec{d_1}=(2,3,4)$ and $\vec{d_2}=(3,7,6)$ (as a combination). $\overrightarrow{AB}$ cannot lie in that plane only if it is NOT a linear combination of $\vec{d_1}$ and $\vec{d_2}$.
$\overrightarrow{AB}$ can be in any direction in the plane of $\vec{d_1},\vec{d_2}$ but NOT $\hat{i}+\hat{j}+2\hat{k}$ since the lines are skew and this vector is not in their common plane.
Correct Answer: 1