Vectors
Coplanarity
MMTS_Full_Test_13
Grade 12
Question:
A, B, C, D are four points with position vectors $\vec{a},\vec{b},\vec{c},\vec{d}$ where $\vec{d}=\alpha\vec{a}+\beta\vec{b}+(1-\alpha-\beta)\vec{c}$. The point D lies on the plane ABC. Which is ALWAYS true?
$\alpha+\beta=1$
$\alpha=\beta$
$|\vec{d}|\ne|\vec{c}|$
$\vec{AD}$ lies in plane ABC
Step-by-Step Solution
Key Concept: If $D$ lies in plane $ABC$, then $\overrightarrow{AD}$ lies in plane $ABC$
$D$ in plane $ABC$ means $\overrightarrow{AD}=\overrightarrow{AB}s+\overrightarrow{AC}t$ for some $s,t$. So $\overrightarrow{AD}$ lies in the plane $ABC$. Option 4 is always true.
Correct Answer: 1