Question

If $|\vec{a}| = 8$, $|\vec{b}| = 5$ and $|\vec{a} - \vec{b}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is equal to

• A. $\dfrac{3\pi}{4}$
• B. $\dfrac{2\pi}{3}$
• C. $\dfrac{\pi}{4}$
• D. $\dfrac{\pi}{6}$
• E. $\dfrac{\pi}{3}$

Hint:

Remember: $|\vec{a}\pm\vec{b}|^2 = |\vec{a}|^2 \pm 2\vec{a}\cdot\vec{b} + |\vec{b}|^2$. This is the vector analogue of the law of cosines and is frequently tested in KEAM.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Use the identity $|\vec{a}-\vec{b}|^2 = |\vec{a}|^2 - 2\vec{a}\cdot\vec{b} + |\vec{b}|^2$ to find $\vec{a}\cdot\vec{b}$, then get the angle.

Step 2: Detailed Explanation:
\[ |\vec{a}-\vec{b}|^2 = 49 = 64 - 2\vec{a}\cdot\vec{b} + 25 = 89 - 2\vec{a}\cdot\vec{b} \] \[ 2\vec{a}\cdot\vec{b} = 89 - 49 = 40 \implies \vec{a}\cdot\vec{b} = 20 \] \[ \cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|} = \frac{20}{8\times 5} = \frac{20}{40} = \frac{1}{2} \] \[ \theta = \cos^{-1}\!\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

Step 3: Final Answer:
The angle between $\vec{a}$ and $\vec{b}$ is $\dfrac{\pi}{3}$.