If → a , → B , → C Are Three Mutually Perpendicular Unit Vectors, Then Prove that ∣ ∣ → a + → B + → C ∣ ∣ = √ 3 - Mathematics

Question

If $\vec{a,} \vec{b,} \vec{c}$ are three mutually perpendicular unit vectors, then prove that $| \vec{a} + \vec{b} + \vec{c} | = \sqrt{3}$

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Solution

$Given that \vec{a}, \vec{b} and \vec{c} are unit vectors .$
$S o, | \vec{a} | = 1, | \vec{b} | = 1 and | \vec{c} | = 1$
$Since they are mutually perpendicular,$
$\vec{a} . \vec{b} = \vec{b} . \vec{c} = \vec{c} . \vec{a} = 0$
$Now,$
${| \vec{a} + \vec{b} + \vec{c} |}^{2} = {| \vec{a} |}^{2} + {| \vec{b} |}^{2} + {| \vec{c} |}^{2} + 2 \vec{a} . \vec{b} + 2 \vec{b} . \vec{c} + 2 \vec{c} . \vec{a}$
$= 1 + 1 + 1 + 0 + 0 + 0$
$= 3$
$∴ | \vec{a} + \vec{b} + \vec{c} | = \sqrt{3}$

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Basic Concepts of Vector Algebra

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Chapter 24: Scalar Or Dot Product - Exercise 24.1 [Page 30]

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Chapter 24 Scalar Or Dot Product
Exercise 24.1 | Q 10 | Page 30

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If → a , → B , → C Are Three Mutually Perpendicular Unit Vectors, Then Prove that ∣ ∣ → a + → B + → C ∣ ∣ = √ 3 - Mathematics

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