Question

The angle in degrees between two vectors \(\vec a=\dfrac{\sqrt3}{2}\hat i+\dfrac{1}{2}\hat j\) and \(\vec b=-\dfrac{\sqrt3}{2}\hat i+\dfrac{1}{2}\hat j\) is

• A. \(30\)
• B. \(60\)
• C. \(90\)
• D. \(120\)

Hint:

Use \(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\) to find the angle between two vectors.

Answer 1

The Correct Option is D

Concept:
The angle \(\theta\) between two vectors \(\vec a\) and \(\vec b\) is given by \[ \vec a\cdot \vec b=|\vec a||\vec b|\cos\theta \] So, \[ \cos\theta=\frac{\vec a\cdot \vec b}{|\vec a||\vec b|} \]

Step 1: Write the given vectors. \[ \vec a=\frac{\sqrt3}{2}\hat i+\frac{1}{2}\hat j \] \[ \vec b=-\frac{\sqrt3}{2}\hat i+\frac{1}{2}\hat j \]

Step 2: Find the dot product. \[ \vec a\cdot \vec b= \left(\frac{\sqrt3}{2}\right)\left(-\frac{\sqrt3}{2}\right) + \left(\frac{1}{2}\right)\left(\frac{1}{2}\right) \] \[ =-\frac{3}{4}+\frac{1}{4} \] \[ =-\frac{2}{4} \] \[ =-\frac{1}{2} \]

Step 3: Find magnitudes. \[ |\vec a|=\sqrt{\left(\frac{\sqrt3}{2}\right)^2+\left(\frac{1}{2}\right)^2} \] \[ =\sqrt{\frac{3}{4}+\frac{1}{4}} \] \[ =1 \] Similarly, \[ |\vec b|=1 \]

Step 4: Find \(\cos\theta\). \[ \cos\theta=\frac{-\frac12}{1\times 1} \] \[ \cos\theta=-\frac12 \]

Step 5: Find \(\theta\). We know that \[ \cos120^\circ=-\frac12 \] Therefore, \[ \theta=120^\circ \]

Step 6: Final answer. \[ \boxed{120^\circ} \]