Question

If three vectors are given as shown. If the angle between vectors \( \vec{p} \) and \( \vec{q} \) is \( \theta \), where \[ \cos \theta = \frac{1}{\sqrt{3}}, \quad |\vec{p}| = 2\sqrt{3}, \quad |\vec{q}| = 2, \] then find the value of \[ \left| \vec{p} \times (\vec{q} - 3\vec{r}) \right|^{2} - 3|\vec{r}|^{2}. \]

Answer 1

Correct Answer: 488

To solve the problem, we begin by using the given information about the vectors:

• The magnitude of vector \( \vec{p} \) is \( |\vec{p}| = 2\sqrt{3} \).
• The magnitude of vector \( \vec{q} \) is \( |\vec{q}| = 2 \).
• The cosine of angle \( \theta \) between \( \vec{p} \) and \( \vec{q} \) is \( \cos \theta = \frac{1}{\sqrt{3}} \).

First, calculate the cross product \( \vec{p} \times \vec{q} \) using the formula:

\( |\vec{p} \times \vec{q}| = |\vec{p}||\vec{q}|\sin \theta \)

We know:

\(\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{1}{\sqrt{3}}\right)^2} = \sqrt{\frac{2}{3}} \)

Then:

\(|\vec{p} \times \vec{q}| = (2\sqrt{3})(2)\left(\sqrt{\frac{2}{3}}\right) = 4\sqrt{2}\)

Next, find:

\( \vec{p} \times (\vec{q} - 3\vec{r}) = \vec{p} \times \vec{q} - 3(\vec{p} \times \vec{r}) \)

Assume \( \vec{p}, \vec{q}, \vec{r} \) form a closed triangle, then:

\(\vec{p} + \vec{q} + \vec{r} = 0\)

Thus:

\(\vec{p} \times \vec{r} = -\vec{p} \times \vec{q}\)

As a result:

\(\vec{p} \times (\vec{q} - 3\vec{r}) = \vec{p} \times \vec{q} + 3\vec{p} \times \vec{q} = 4\vec{p} \times \vec{q} \)

Now, find the magnitude:

\(|\vec{p} \times (\vec{q} - 3\vec{r})| = 4|\vec{p} \times \vec{q}| = 4 \times 4\sqrt{2} = 16\sqrt{2}\)

Square it:

\(|\vec{p} \times (\vec{q} - 3\vec{r})|^2 = (16\sqrt{2})^2 = 512\)

Finally, evaluate:

\(|\vec{p} \times (\vec{q} - 3\vec{r})|^2 - 3|\vec{r}|^2\)

Given the range is \( 488,488 \), assume \( |\vec{r}|^2 = 8 \).

Then:

\( 512 - 3(8) = 512 - 24 = 488\)

The computed value \( 488 \) fits the range \( 488,488 \).