Vector Algebra Question 22
Question 22  01 February  Shift 2
Let $\vec{a}=5 \hat{i}\hat{j}3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$ be two vectors. Then which one of the following statements is TRUE ?

Projection of $\vec{a}$ on $\vec{b}$ is $\frac{13}{\sqrt{35}}$ and the direction of the projection vector is same as $\vec{b}$.

Projection of $\vec{a}$ on $\vec{b}$ is $\frac{13}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$.

Projection of $\vec{a}$ on $\vec{b}$ is $\frac{13}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$.

Projection of $\vec{a}$ on $\vec{b}$ is $\frac{13}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$.
Show Answer
Answer: (1)
Solution:
Formula: Scalar product of two vectors: projection of a vector on the another vector
Projection of $\vec{a}$ 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{5315}{\sqrt{35}}$
$=\frac{13}{\sqrt{35}}$
Negative sign indicates that direction of the projection vector is opposite to the direction of $\vec{b}$.