If a- b- and c are three vectors such that -a-3- -b-5- b-c-10- and the angle between b and c is -3- If a is perpendicular to b-c- then -a-b-c-

Step 1. Find the value of |c|.It is given that b·c=10 , |b|=5 and the angle between b and c is π/3.So using the relation of dot product b·c=|b||c|cos(b^c) the v ...

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Byju's Answer

Standard XII

Mathematics

Angle between Two Lines

If a, b, and ...

Question

If $\vec{a}, \vec{b},$ and $\vec{c}$ are 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 $\frac{π}{3}$ . If $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$ , then $|\vec{a} \times (\vec{b} \times \vec{c})|$

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Solution

Step 1. Find the value of $|\vec{c}|$ .
It is given that $\vec{b} \cdot \vec{c} = 10, |\vec{b}| = 5$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{π}{3}$ .
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:
$\vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos (\vec{b}^\vec{c}) \Rightarrow 10 = 5 |\vec{c}| \cos \frac{π}{3} \Rightarrow 2 = |\vec{c}| \times \frac{1}{2} [\cos \frac{π}{3} = \frac{1}{2}] \Rightarrow 4 = |\vec{c}|$
Step 2. Find the value of $|\vec{b} \times \vec{c}|$ .
The value of the cross product can be found as: $|\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin (\vec{b}^\vec{c})$ .
Now, $|\vec{b}| = 5, |\vec{c}| = 4$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{π}{3}$ , so the value of $|\vec{b} \times \vec{c}|$ is obtained as:
$\begin{array}{rcl} |\vec{b} \times \vec{c}| & = & |\vec{b}| |\vec{c}| \sin (\vec{b}^\vec{c}) \\ = & 5 \times 4 \times \sin (\frac{π}{3}) \\ = & 20 \times \frac{\sqrt{3}}{2} [\sin \frac{π}{3} = \frac{\sqrt{3}}{2}] \\ = & 10 \sqrt{3} \end{array}$
Step 3. Find the value of $|\vec{a} \times (\vec{b} \times \vec{c})|$ .
The value of the cross product can be found as: $|\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}| |\vec{b} \times \vec{c}| \sin (\vec{a}^(\vec{b} \times \vec{c}))$
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 $\frac{π}{2}$ , so the value of $|\vec{a} \times (\vec{b} \times \vec{c})|$ is obtained as:
$\begin{array}{rcl} |\vec{a} \times (\vec{b} \times \vec{c})| & = & |\vec{a}| |\vec{b} \times \vec{c}| \sin (\vec{a}^(\vec{b} \times \vec{c})) \\ = & \sqrt{3} \times 10 \sqrt{3} \times \sin \frac{π}{2} \\ = & 30 \times 1 [\sin \frac{π}{2} = 1] \\ = & 30 \end{array}$
So, for the given condition the value of $|\vec{a} \times (\vec{b} \times \vec{c})|$ is $30$ .
Hence, the answer is $30$ .

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If a→,b→, and c→ are three vectors such that a→=3,b→=5,b→·c→=10, and the angle between b→ and c→ is π3. If a→ is perpendicular to b→×c→, then a→×b→×c→