To find the value of $$|\bar{a} \times (\bar{b} \times \bar{c})|$$, we need to utilize given information step by step. First, let's understand the vectors and their relationships stated in the problem.

Given:

$$|\vec{a}| = \sqrt{3}$$
$$|\vec{b}| = 5$$
$$ \vec{b} \cdot \vec{c} = 10$$
The angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\frac{\pi}{3}$$
$$\vec{a}$$ is perpendicular to $$\vec{b} \times \vec{c}$$

To find $$|\vec{b} \cdot \vec{c}|$$ given the angle between them and the magnitude of $$\vec{b}$$, we apply the formula for dot product in terms of the angle between the vectors:

$ \vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos(\theta) $

Substituting the known values, we get:

$ 10 = 5 |\vec{c}| \cos\left(\frac{\pi}{3}\right) $

Since $$\cos(\frac{\pi}{3}) = \frac{1}{2}$$, we have:

$ 10 = 5 |\vec{c}| \frac{1}{2} $

$ |\vec{c}| = 4 $

So the magnitude of $$\vec{c}$$ is 4.

Given that $$\vec{a}$$ is perpendicular to $$\vec{b} \times \vec{c}$$, we can find the magnitude of the cross product $$|\vec{b} \times \vec{c}|$$ using the formula for the cross product in terms of magnitudes and angles:

$ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin(\theta) $

Substituting the known values, we get:

$ |\vec{b} \times \vec{c}| = 5 \cdot 4 \cdot \sin\left(\frac{\pi}{3}\right) $

Since $$\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$$, we have:

$ |\vec{b} \times \vec{c}| = 20 \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3} $

Now, to find $$|\vec{a} \times (\vec{b} \times \vec{c})|$$, we use the fact that the magnitude of the cross product of two vectors is given by the product of their magnitudes and the sine of the angle between them. Since $$\vec{a}$$ is perpendicular to $$\vec{b} \times \vec{c}$$, the angle between them is $$\frac{\pi}{2}$$, and $$\sin\left(\frac{\pi}{2}\right) = 1$$. Therefore, the magnitude of their cross product is:

$ |\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}| |\vec{b} \times \vec{c}| \sin\left(\frac{\pi}{2}\right) $

$ = \sqrt{3} \cdot 10\sqrt{3} \cdot 1 = 3 \cdot 10 = 30 $

Thus, the correct answer is:

Option B: 30

Vector Algebra

Continue this paper

Similar Vector Algebra PYQs

More practice from this question