Vectors
Cross product and angle between vectors
MJAT_TS1_P2
Grade 12
Question:
Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}| = 1$, $|\vec{b}| = 4$, $\vec{a}\cdot\vec{b} = 2$. If $\vec{c} = 2(\vec{a}\times\vec{b}) - 3\vec{b}$, then which of the following is/are correct?
A) $\vec{b}\cdot\vec{c} = 48$
B) $\vec{b}\cdot\vec{c} = -48$
C) Angle between $\vec{b}$ and $\vec{c}$ is $\dfrac{5\pi}{6}$
D) Angle between $\vec{b}$ and $\vec{c}$ is $\dfrac{\pi}{6}$
Step-by-Step Solution
Key Concept: $\vec{b}\cdot(\vec{a}\times\vec{b}) = 0$ always. So $\vec{b}\cdot\vec{c} = 2\vec{b}\cdot(\vec{a}\times\vec{b}) - 3|\vec{b}|^2 = 0 - 3\cdot 16 = -48$.
$\vec{b}\cdot\vec{c} = -48$ (B ✓). $|\vec{c}| = \sqrt{192} = 8\sqrt{3}$. $\cos\alpha = \frac{\vec{b}\cdot\vec{c}}{|\vec{b}||\vec{c}|} = \frac{-48}{4\cdot 8\sqrt{3}} = \frac{-48}{32\sqrt{3}} = -\frac{\sqrt{3}}{2}$. So angle $= \frac{5\pi}{6}$ (C ✓).
Correct Answer: BC