Vectors & 3D Geometry
Unit vectors; constraint to p+q+r=0
MMTS_Full_Test_20
Grade 12
Question:
Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to
(A) 0
(B) 3
(C) 5
(D) $\dfrac{3}{2}$
Step-by-Step Solution
Key Concept: $|\vec{p}-\vec{q}|^2+\cdots=2(3)-2(\vec{p}\cdot\vec{q}+\vec{q}\cdot\vec{r}+\vec{r}\cdot\vec{p})=9 \Rightarrow \vec{p}\cdot\vec{q}+\vec{q}\cdot\vec{r}+\vec{r}\cdot\vec{p}=-3/2$.
$|2\vec{p}+5\vec{q}+5\vec{r}|=3$.
Correct Answer: (B) 3