Question

If $|\vec{a}| = \sqrt{3}$; $|\vec{b}| = 5$; $\vec{b} \cdot \vec{c} = 10$, angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$, $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$. Then the value of $|\vec{a} \times (\vec{b} \times \vec{c})|$ is

• A. 20
• B. 30
• C. 60
• D. 40

Hint:

Remember the formulas for dot product and cross product, and how to use the condition of perpendicularity for the angle between vectors. If $\vec{u}$ is perpendicular to $\vec{v}$, then the angle between them is $\frac{\pi}{2}$, and $\sin(\frac{\pi}{2}) = 1$.

Answer 1

The Correct Option is A

Step 1: Find the magnitude of $\vec{c}$. \[ \vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos \theta \] Given $\vec{b} \cdot \vec{c} = 10$, $|\vec{b}| = 5$, and $\theta = \frac{\pi}{3}$. \[ 10 = 5 \cdot |\vec{c}| \cdot \cos\left(\frac{\pi}{3}\right) \] \[ 10 = 5 \cdot |\vec{c}| \cdot \frac{1}{2} \] \[ 10 = \frac{5}{2} |\vec{c}| \] \[ |\vec{c}| = \frac{10 \cdot 2}{5} = 4 \]
Step 2: Find the magnitude of the cross product $|\vec{b} \times \vec{c}|$. \[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin \theta \] \[ |\vec{b} \times \vec{c}| = 5 \cdot 4 \cdot \sin\left(\frac{\pi}{3}\right) \] \[ |\vec{b} \times \vec{c}| = 20 \cdot \frac{\sqrt{3{2} \] \[ |\vec{b} \times \vec{c}| = 10\sqrt{3} \]
Step 3: Calculate $|\vec{a} \times (\vec{b} \times \vec{c})|$.We are given that $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$.Let $\vec{X} = \vec{b} \times \vec{c}$. Then $\vec{a}$ is perpendicular to $\vec{X}$.The angle between $\vec{a}$ and $\vec{X}$ is $\phi = \frac{\pi}{2}$. \[ |\vec{a} \times \vec{X}| = |\vec{a}| |\vec{X}| \sin \phi \] \[ |\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}| |\vec{b} \times \vec{c}| \sin\left(\frac{\pi}{2}\right) \] Given $|\vec{a}| = \sqrt{3}$. \[ |\vec{a} \times (\vec{b} \times \vec{c})| = \sqrt{3} \cdot 10\sqrt{3} \cdot 1 \] \[ |\vec{a} \times (\vec{b} \times \vec{c})| = 3 \cdot 10 \] \[ |\vec{a} \times (\vec{b} \times \vec{c})| = 30 \]