JEE Mathematics 2023 Question with Solution

Given: $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$.

Find: The scalar projection of $\vec{a}$ on $\vec{b}$ and the corresponding correct option.

The scalar projection of $\vec{a}$ on $\vec{b}$ is

$$\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$$

Now,

$$\vec{a} \cdot \vec{b} = (5)(1) + (-1)(3) + (-3)(5) = 5 - 3 - 15 = -13$$

and

$$|\vec{b}| = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{35}$$

Therefore, the projection is

$$\frac{-13}{\sqrt{35}}$$

Since the value is negative, the projection vector is opposite in direction to $\vec{b}$.

The solution explicitly states that the correct computed projection is $\frac{-13}{\sqrt{35}}$, but no option matches this value and the answer key says Drop. Among the listed options, B is the most defensible because it has the correct negative sign and opposite direction, though the magnitude shown in the option is inconsistent with the working.

Therefore, the closest matching option is B.

Given: $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$.

Find: Whether any listed statement matches the actual projection.

From the working shown in the solution,

$$\vec{a} \cdot \vec{b} = -13$$

so the scalar projection is

$$\frac{-13}{\sqrt{35}}$$

This means two things:

1. The projection is negative.
2. Its direction is opposite to $\vec{b}$.

Hence, any correct option must contain a negative value and state opposite direction. The solution also says the question is marked as dropped because no option matches the computed value exactly.

Therefore, the options are inconsistent with the actual calculation, and B is chosen only as the nearest logical match.