Question

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \( |\vec{a}|=\sqrt{3} \), \( |\vec{b}|=5 \), \( \vec{b}\cdot\vec{c}=10 \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{3} \). If \( \vec{a} \) is perpendicular to the vector \( \vec{b}\times\vec{c} \), then \( |\vec{a}\times(\vec{b}\times\vec{c})| \) is equal to

• A. $10\sqrt{3}$
• B. $5\sqrt{3}$
• C. 60
• D. 30

Hint:

Vector Tip:The magnitude of a cross product $|\overline{A}\times\overline{B}|$ gives the area of the parallelogram formed by vectors A and B, defined mathematically as $|\overline{A}||\overline{B}|\sin\theta$. Always treat parenthesis grouped vectors like $(\overline{b}\times\overline{c})$ as a single distinct vector when calculating subsequent products.

Answer 1

The Correct Option is D

Concept: Vector Algebra - Dot and Cross Products.

Step 1: Use the dot product to find the magnitude of vector $\overline{c}$. We are given $\overline{b}\cdot\overline{c}=10$. Using the formula for the dot product, $|\overline{b}||\overline{c}|\cos\theta = 10$. Substituting the known values ($|\overline{b}|=5$ and $\theta = \frac{\pi}{3}$), we get $5 \cdot |\overline{c}| \cdot \cos(\frac{\pi}{3}) = 10$.

Step 2: Solve for $|\overline{c}|$. Since $\cos(\frac{\pi}{3}) = \frac{1}{2}$, the equation becomes $5 \cdot |\overline{c}| \cdot \frac{1}{2} = 10$. Multiplying both sides by $\frac{2}{5}$ gives us $|\overline{c}| = 4$.

Step 3: Set up the magnitude of the target cross product. We need to evaluate $|\overline{a}\times(\overline{b}\times\overline{c})|$. Let $\overline{v} = \overline{b}\times\overline{c}$. The magnitude of a cross product is $|\overline{a}\times\overline{v}| = |\overline{a}||\overline{v}|\sin\alpha$, where $\alpha$ is the angle between $\overline{a}$ and $\overline{v}$.

Step 4: Apply the perpendicularity condition. The problem states that $\overline{a}$ is perpendicular to $\overline{b}\times\overline{c}$. Therefore, the angle $\alpha = \frac{\pi}{2}$, and $\sin(\frac{\pi}{2}) = 1$. The expression simplifies to $|\overline{a}||\overline{b}\times\overline{c}|$.

Step 5: Calculate the final value. Expand $|\overline{b}\times\overline{c}|$ as $|\overline{b}||\overline{c}|\sin(\frac{\pi}{3})$. Substitute all known magnitudes: $|\overline{a}| \cdot (|\overline{b}||\overline{c}|\sin\frac{\pi}{3}) = \sqrt{3} \cdot (5 \cdot 4 \cdot \frac{\sqrt{3}}{2})$. This simplifies to $\sqrt{3} \cdot 20 \cdot \frac{\sqrt{3}}{2} = 10 \cdot 3 = 30$. $$ \therefore \text{The value of } |\overline{a}\times(\overline{b}\times\overline{c})| \text{ is } 30. $$