If $|\overline{\mathrm{a}}|=\sqrt{3} ;|\overline{\mathrm{b}}|=5 ; \overline{\mathrm{b}} \cdot…
If $|\overline{\mathrm{a}}|=\sqrt{3} ;|\overline{\mathrm{b}}|=5 ; \overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=10$, angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $\frac{\pi}{3}, \overline{\mathrm{a}}$ is perpendicular to $\overline{\mathrm{b}} \times \overline{\mathrm{c}}$. Then the value of $|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|$ is
$20$
$30$
$60$
$40$
Solution
Step 1. Find the value of \(|\overrightarrow{c}|\).
It is given that \(\vec{b} \cdot \vec{c}=10,|\vec{b}|=5\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\frac{\pi}{3}\).
So using the relation of dot product \(\vec{b} \cdot \vec{c}=|\vec{b}| \vec{c} | \cos (\vec{b} \vec{c})\) the value of \(|\vec{c}|\) can be obtained as:
\(\begin{aligned}
& \vec{b} \cdot \vec{c}=|\vec{b}| \vec{c} \mid \cos (\vec{b}, \vec{c}) \\
& \Rightarrow 10=5|\vec{c}| \cos \frac{\pi}{3} \\
& \Rightarrow 2=|\vec{c}| \times \frac{1}{2} \left[\cos \frac{\pi}{3}=\frac{1}{2}\right] \\
& \Rightarrow 4=|\vec{c}|
\end{aligned}\)
Step 2. Find the value of \(\vec{b} \times \vec{c} \mid\).
The value of the cross product can be found as: \(|\vec{b} \times \overrightarrow{c}|=|\vec{b}||\vec{c}| \sin (\vec{b}, \vec{c})\).
Now, \(|\vec{b}|=5,|\vec{c}|=4\) and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\frac{\pi}{3}\), so the value of \(|\vec{b} \times \vec{c}|\) is obtained as:
\(\begin{aligned}
|\vec{b} \times \vec{c}| & =|\vec{b}||\vec{c}| \sin (\vec{b}, \vec{c}) \\
& =5 \times 4 \times \sin \left(\frac{\pi}{3}\right) \\
& =20 \times \frac{\sqrt{3}}{2} \left[\sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}\right] \\
& =10 \sqrt{3}
\end{aligned}\)
Step 3. Find the value of \(|\vec{a} \times(\vec{b} \times \vec{c})|\).
The value of the cross product can be found as:
\(|\vec{a} \times(\vec{b} \times \vec{c})|=|\vec{a}||\vec{b} \times \vec{c}| \sin (\vec{a} \times(\vec{b} \times \vec{c}))\)
Now, \(|\vec{a}|=\sqrt{3},|\vec{b} \times \vec{c}|=10 \sqrt{3}\) and the angle between \(\vec{a}\) and \(\vec{b} \times \vec{c}\) is \(\frac{\pi}{2}\), so the value of \(|\vec{a} \times(\vec{b} \times \vec{c})|\) is obtained as:
\(\begin{aligned}
|\vec{a} \times(\vec{b} \times \vec{c})| & =|\vec{a}||\vec{b} \times \vec{c}| \sin (\vec{a} \cdot(\vec{b} \times \vec{c})) \\
& =\sqrt{3} \times 10 \sqrt{3} \times \sin \frac{\pi}{2} \\
& =30 \times 1 \left[\sin \frac{\pi}{2}=1\right] \\
& =30
\end{aligned}\)
So, for the given condition the value of \(|\vec{a} \times(\vec{b} \times \vec{c})|\) is 30.
Hence, the answer is 30.