JEE MAIN - Mathematics (2020 - 2nd September Morning Slot - No. 10)
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$$ = 8.
Then $${\left| {\overrightarrow a + 2\overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}$$ is equal to ______.
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$$ = 8.
Then $${\left| {\overrightarrow a + 2\overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}$$ is equal to ______.
Answer
2
Explanation
Given, $$\left| {\overrightarrow a } \right| = \left| {\overrightarrow b } \right| = \left| {\overrightarrow c } \right| = 1$$
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$$ = 8
$$ \Rightarrow $$ $${\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow b } \right|^2} - 2\overrightarrow a .\overrightarrow b + {\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow c } \right|^2} - 2\overrightarrow a .\overrightarrow c $$ = 8
$$ \Rightarrow $$ $$\overrightarrow a .\overrightarrow b + \overrightarrow a .\overrightarrow c $$ = -2
Now, $${\left| {\overrightarrow a + 2\overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}$$
= $${\left| {\overrightarrow a } \right|^2} + 4{\left| {\overrightarrow b } \right|^2} + 4\overrightarrow a .\overrightarrow b + {\left| {\overrightarrow a } \right|^2} + 4{\left| {\overrightarrow c } \right|^2} + 4\overrightarrow a .\overrightarrow c $$
= 10 + 4$$\left( {\overrightarrow a .\overrightarrow b + \overrightarrow a .\overrightarrow c } \right)$$
= 10 + 4(-2)
= 2
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$$ = 8
$$ \Rightarrow $$ $${\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow b } \right|^2} - 2\overrightarrow a .\overrightarrow b + {\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow c } \right|^2} - 2\overrightarrow a .\overrightarrow c $$ = 8
$$ \Rightarrow $$ $$\overrightarrow a .\overrightarrow b + \overrightarrow a .\overrightarrow c $$ = -2
Now, $${\left| {\overrightarrow a + 2\overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}$$
= $${\left| {\overrightarrow a } \right|^2} + 4{\left| {\overrightarrow b } \right|^2} + 4\overrightarrow a .\overrightarrow b + {\left| {\overrightarrow a } \right|^2} + 4{\left| {\overrightarrow c } \right|^2} + 4\overrightarrow a .\overrightarrow c $$
= 10 + 4$$\left( {\overrightarrow a .\overrightarrow b + \overrightarrow a .\overrightarrow c } \right)$$
= 10 + 4(-2)
= 2
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