Question

if a,b,c be three unit vector sch that ax(bxc)=b/2.
Find the angle which a makes with b & c , B & c being non- parallel.

Answer 1

Answer on Question #40414 – Math – Analytic Geometry

If $a, b, c$ be three unit vector, sch that $ax(bxc) = b/2$. Find the angle which $a$ makes with $b \& c$, $B \& c$ being non-parallel.

Solution.

Let angles between $\vec{a}$ and $\vec{b}$ and between $\vec{a}$ and $\vec{c}$ be $\alpha$ and $\beta$ respectively.

We have,

$$\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b}}{2}$$

By property $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \cdot \vec{b} - (\vec{a} \cdot \vec{b}) \cdot \vec{c}$:

$$(\vec{a} \cdot \vec{c}) \cdot \vec{b} - (\vec{a} \cdot \vec{b}) \cdot \vec{c} = \frac{\vec{b}}{2}$$ $$(\vec{a} \cdot \vec{c} - \frac{1}{2}) \cdot \vec{b} - (\vec{a} \cdot \vec{b}) \cdot \vec{c} = 0$$

or

$$\vec{a} \cdot \vec{c} = \frac{1}{2} \quad \text{and} \quad \vec{a} \cdot \vec{b} = 0 \quad \text{(As } \vec{b} \text{ and } \vec{c} \text{ are non-parallel)}$$

Then the angles can be found:

$$\cos \beta = \frac{1}{2}, \quad \cos \alpha = 0$$

or

$$\beta = \frac{\pi}{3}, \quad \alpha = \frac{\pi}{2}.$$

**Answer**: angle between $\vec{a}$ and $\vec{b}$ is $\pi/3$, angle between $\vec{a}$ and $\vec{c}$ is $\pi/2$.