JEE MAIN - Mathematics (2020 - 9th January Evening Slot - No. 9)
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\left| {\overrightarrow a } \right| = \sqrt 3 $$,
$$\left| {\overrightarrow b } \right| = 5,\overrightarrow b .\overrightarrow c = 10$$ and the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$
is $${\pi \over 3}$$. If $${\overrightarrow a }$$ is perpendicular to the vector $$\overrightarrow b \times \overrightarrow c $$ , then $$\left| {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)} \right|$$ is equal to _____.
Answer
30
Explanation
Given $$\left| {\overrightarrow a } \right| = \sqrt 3 $$,
$$\left| {\overrightarrow b } \right| = 5$$
Given $$\overrightarrow b .\overrightarrow c = 10$$
And the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$ is $${\pi \over 3}$$
$$ \therefore $$ $$bc\cos {\pi \over 3}$$ = 10
$$ \Rightarrow $$ c = 4
$${\overrightarrow a }$$ is perpendicular to the vector $$\overrightarrow b \times \overrightarrow c $$
$$ \therefore $$ $$\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$$ = 0 and angle between them is $${\pi \over 2}$$
Now $$\left| {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)} \right|$$
= $$\left| {\overrightarrow a } \right|\left| {\overrightarrow b \times \overrightarrow c } \right|\sin {\pi \over 2}$$
= $$\left| {\overrightarrow a } \right|$$.$${\left| {\overrightarrow b } \right|.\left| {\overrightarrow c } \right|}$$$$\sin {\pi \over 3}$$.1
= $$\sqrt 3 \times 5 \times 4 \times {{\sqrt 3 } \over 2}$$
= 30
Given $$\overrightarrow b .\overrightarrow c = 10$$
And the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$ is $${\pi \over 3}$$
$$ \therefore $$ $$bc\cos {\pi \over 3}$$ = 10
$$ \Rightarrow $$ c = 4
$${\overrightarrow a }$$ is perpendicular to the vector $$\overrightarrow b \times \overrightarrow c $$
$$ \therefore $$ $$\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$$ = 0 and angle between them is $${\pi \over 2}$$
Now $$\left| {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)} \right|$$
= $$\left| {\overrightarrow a } \right|\left| {\overrightarrow b \times \overrightarrow c } \right|\sin {\pi \over 2}$$
= $$\left| {\overrightarrow a } \right|$$.$${\left| {\overrightarrow b } \right|.\left| {\overrightarrow c } \right|}$$$$\sin {\pi \over 3}$$.1
= $$\sqrt 3 \times 5 \times 4 \times {{\sqrt 3 } \over 2}$$
= 30
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