Verified Solution • 2025–26 Session moderate

QJEEM23Ma3043: JEE Main 2023 Physics - Class 11 JEEM23Ma3043 Solution: Let and be two vectors. Then which one of the following…

Let a⃗=5i^−j^−3k^\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k} and b⃗=i^+3j^+5k^\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k} be two vectors. Then which one of the following statements is TRUE ?

Multiple Choice Options

  • A Projection of a⃗\vec{a} on b⃗\vec{b} is −1335\frac{-13}{\sqrt{35}} and the direction of the projection vector is opposite to the direction of b⃗\vec{b} .
  • B Projection of a⃗\vec{a} on b⃗\vec{b} is 1335\frac{13}{\sqrt{35}} and the direction of the projection vector is opposite to the direction of b⃗\vec{b} .
  • C Projection of a⃗\vec{a} on b⃗\vec{b} is 1335\frac{13}{\sqrt{35}} and the direction of the projection vector is same as of b⃗\vec{b} .
  • D Projection of a⃗\vec{a} on b⃗\vec{b} is −1335\frac{-13}{\sqrt{35}} and the direction of the projection vector is same as of b⃗\vec{b} .

Step-by-Step Verified Solution

amp; Projection of a⃗ on b⃗=a⃗⋅b⃗∣b⃗∣amp;=(5i^−j^−3k^)⋅(i^+3j^+5k^)12+32+52=5−3−1535amp;=−1335\begin{aligned} & \text { Projection of }\vec{a} \text { on } \vec{b} =\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \\\\ & = \frac{(5 \hat{i}-\hat{j}-3 \hat{k}) \cdot(\hat{i}+3 \hat{j}+5 \hat{k})}{\sqrt{1^2+3^2+5^2}}=\frac{5-3-15}{\sqrt{35}} \\\\ & = \frac{-13}{\sqrt{35}}\end{aligned}

Negative sign indicates that direction of the projection vector is opposite to the direction of b⃗\vec{b} .

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