Question

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

• A. $10\sqrt{3}$
• B. 15
• C. 30
• D. 10

Answer 1

Answer: (C)

30

Answer 2

Solution:

We are given:

• $|a| = \sqrt{3}, |b| = 5$
• The dot product $b \cdot c = 10$
• The angle between $b$ and $c$ is $\frac{\pi}{3}$.

Since $b \cdot c = |b||c| \cos(\frac{\pi}{3})$, we have:

$10 = 5 \cdot |c| \cdot (\frac{1}{2})$
$|c| = \frac{10 \times 2}{5} = 4$

Compute $|b \times c|$ using:

$|b \times c| = |b||c| \sin(\frac{\pi}{3}) = 5 \cdot 4 \cdot (\frac{\sqrt{3}}{2}) = 10\sqrt{3}$

Given $a \perp (b \times c)$, we have:

$|a \times (b \times c)| = |a| \cdot |b \times c| = \sqrt{3} \cdot 10\sqrt{3} = 30$

Answer: Option (c) 30