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If $\vec{a}, \vec{b} a n d \vec{c}$ are three mutually perpendicular unit vectors and $\vec{d}$ is a unit vector which makes equal angles with $\vec{a}, \vec{b} a n d \vec{c}$ then the value of ${| \vec{a} + \vec{b} + \vec{c} + \vec{d} |}^{2}$

Option 1 -

4 + $2 \sqrt[]{2}$

Option 2 -

4 + $2 \sqrt[]{3}$

Option 3 -

2 + $\sqrt[]{5}$

Option 4 -

3 + $\sqrt[]{5}$

2 Views | Posted 3 weeks ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

3 weeks ago

Correct Option - 2

Detailed Solution:

$| \vec{a} + \vec{b} + \vec{c} + \vec{d} | = \sum {| \vec{a} |}^{2} + 2 \sum \vec{a} . \vec{b}$

$= 4 + 2 \vec{d} . (\vec{a} + \vec{b} + \vec{c})$ (a, b, c are mutually ^ r)

Let $\vec{d} = λ \vec{a} + μ \vec{b} + v \vec{c}$

Also $λ^{2} + μ^{2} + v^{2} = 1 = 3 c o s^{2} θ o r c o s θ = \frac{1}{\sqrt[]{3}}$

$∴ {| \vec{a} + \vec{b} + \vec{c} + \vec{d} |}^{2} = 4 \pm 2 . \frac{3}{\sqrt[]{3}}$

$= 4 \pm 2 \frac{3}{\sqrt[]{3}}$

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If $\vec{a}, \vec{b} a n d \vec{c}$ are three mutually perpendicular unit vectors and $\vec{d}$ is a unit vector which makes equal angles with $\vec{a}, \vec{b} a n d \vec{c}$ then the value of ${| \vec{a} + \vec{b} + \vec{c} + \vec{d} |}^{2}$

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If a → , b → a n d c → are three mutually perpendicular unit vectors and d → is a unit vector which makes equal angles with a → , b → a n d c → then the value of | a → + b → + c → + d → | 2

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If $\vec{a}, \vec{b} a n d \vec{c}$ are three mutually perpendicular unit vectors and $\vec{d}$ is a unit vector which makes equal angles with $\vec{a}, \vec{b} a n d \vec{c}$ then the value of ${| \vec{a} + \vec{b} + \vec{c} + \vec{d} |}^{2}$