Let → a = 5 ^ i - ^ j - 3 ^ k and → b = ^ i + 3 ^ j + 5 ^ k be two vectors. Then which one of the following statements is TRUE?

Correct answer: A. Projection of → a on → b is - 13 35 and the direction of the projection vector is opposite to the direction of → b Given → a = 5 ^ i - ^ j - 3 ^ k So, | → a | = 5 ^ 2 + 1 ^ 2 + 3 ^ 2 = 35 → b = ^ i + 3 ^ j + 5 ^ k So, 5 ^ 2 + 1 ^ 2 + 3 ^ 2 = 35 So, → a · → b = 5 - 3 - 15 = - 13 Now finding the projection of → a on → b using the formula we get, → a · → b | → b | = - 13 35 And projection of → b on → a is given by, → a · → b | → a | = - 13 35 Note- The original answer key was dropped by NTA in Jee mains 1st feb 2023, here we have modified the options.

JEE Main 20231 FebruaryEvening Shift

Exam: JEE Main
Year: 2023
Date: 1 February
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2023 · 1 February · Evening Shift · Mathematics · Vector Algebra

Let → a = 5 ^ i - ^ j - 3 ^ k and → b = ^ i + 3 ^ j + 5 ^ k be two...

Let

\overset{\to}{a} = 5\overset{^}{i} - \overset{^}{j} - 3\overset{^}{k}

and

\overset{\to}{b} = \overset{^}{i} + 3\overset{^}{j} + 5\overset{^}{k}

be two vectors. Then which one of the following statements is TRUE?

Options

AProjection of $\overset{\to}{a}$ on $\overset{\to}{b}$ is $\dfrac{- 13}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\overset{\to}{b}$
BProjection of $\overset{\to}{a}$ on $\overset{\to}{b}$ is $\dfrac{- 17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\overset{\to}{b}$
CProjection of $\overset{\to}{a}$ on $\overset{\to}{b}$ is $\dfrac{17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\overset{\to}{b}$
DProjection of $\overset{\to}{a}$ on $\overset{\to}{b}$ is $\dfrac{13}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\overset{\to}{a}$

Correct answer

A. Projection of $\overset{\to}{a}$ on $\overset{\to}{b}$ is $\dfrac{- 13}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\overset{\to}{b}$

Step-by-step solution

Given

$\overset{\to}{a} = 5\overset{^}{i} - \overset{^}{j} - 3\overset{^}{k}$

So, \left|\overset{\to}{a}\right| = \sqrt{{5}^{2} + {1}^{2} + {3}^{2}} = \sqrt{35} $$ \overset{\to}{b} = \overset{^}{i} + 3\overset{^}{j} + 5\overset{^}{k} $So,$ \sqrt{{5}^{2} + {1}^{2} + {3}^{2}} = \sqrt{35} $So,$ \overset{\to}{a} \cdot \overset{\to}{b} = 5 - 3 - 15 = - 13 $Now finding the projection of$ \overset{\to}{a} $on$ \overset{\to}{b} $using the formula we get,$ \dfrac{\overset{\to}{a} \cdot \overset{\to}{b}}{\left|\overset{\to}{b}\right|} = - \dfrac{13}{\sqrt{35}} $And projection of$ \overset{\to}{b} $on$ \overset{\to}{a} $is given by,$ \dfrac{\overset{\to}{a} \cdot \overset{\to}{b}}{\left|\overset{\to}{a}\right|} = - \dfrac{13}{\sqrt{35}}$

Note- The original answer key was dropped by NTA in Jee mains 1st feb 2023, here we have modified the options.

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