Question

If angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\pi/3$ then angle between $\overrightarrow{a}$ and $-2\overrightarrow{b}$ is :

• A. $\pi/4$
• B. $2\pi/3$
• C. $\pi$
• D. $\pi/3$

Answer 1

Answer: (B)

$2\pi/3$

Answer 2

Let $\theta$ be the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. We are given $\theta = \pi/3$. Let $\phi$ be the angle between vectors $\overrightarrow{a}$ and $-2\overrightarrow{b}$. We want to find $\phi$.

The angle between two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ can be found using the dot product formula: $\cos \theta = \frac{\overrightarrow{u} \cdot \overrightarrow{v}}{|\overrightarrow{u}| |\overrightarrow{v}|}$

For the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$, we have: $\cos(\pi/3) = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{a}| |\overrightarrow{b}|}$

Since $\cos(\pi/3) = 1/2$, we have $\frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{a}| |\overrightarrow{b}|} = 1/2$.

For the angle $\phi$ between $\overrightarrow{a}$ and $-2\overrightarrow{b}$, we have: $\cos \phi = \frac{\overrightarrow{a} \cdot (-2\overrightarrow{b})}{|\overrightarrow{a}| |-2\overrightarrow{b}|}$

Using the properties of the dot product and vector magnitude: $\overrightarrow{a} \cdot (-2\overrightarrow{b}) = -2 (\overrightarrow{a} \cdot \overrightarrow{b})$ $|-2\overrightarrow{b}| = |-2| |\overrightarrow{b}| = 2 |\overrightarrow{b}|$

Substitute these into the formula for $\cos \phi$: $\cos \phi = \frac{-2 (\overrightarrow{a} \cdot \overrightarrow{b})}{|\overrightarrow{a}| (2 |\overrightarrow{b}|)}$ $\cos \phi = \frac{- (\overrightarrow{a} \cdot \overrightarrow{b})}{|\overrightarrow{a}| |\overrightarrow{b}|}$

We know that $\frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{a}| |\overrightarrow{b}|} = \cos(\pi/3) = 1/2$. So, $\cos \phi = - (1/2) = -1/2$.

The angle $\phi$ between two vectors is conventionally taken to be in the range $[0, \pi]$. We need to find $\phi \in [0, \pi]$ such that $\cos \phi = -1/2$. The angle is $\phi = 2\pi/3$.