1. Given vectors

   $\vec a = 5\hat i - \hat j - 3\hat k, \qquad \vec b = \hat i + 3\hat j + 5\hat k$

2. Find the dot product $\vec a \cdot \vec b$

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

3. Find the magnitude of $\vec b$

   $|\vec b| = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{1+9+25} = \sqrt{35}$

4. Scalar projection of $\vec a$ on $\vec b$

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

   $\text{proj}_{\vec b}(\vec a) = \frac{\vec a \cdot \vec b}{|\vec b|} = \frac{-13}{\sqrt{35}}$

5. Direction of the projection vector

   The vector projection is

   $\text{vector projection of } \vec a \text{ on } \vec b = \frac{\vec a \cdot \vec b}{|\vec b|^2}\,\vec b$

   Since

   $\vec a \cdot \vec b = -13 < 0$

   the coefficient of $\vec b$ is negative. Therefore, the projection vector is in the opposite direction to $\vec b$.

6. Check options

   • A: $\dfrac{-13}{\sqrt{35}}$ and opposite direction to $\vec b$ ✅
   • B: magnitude sign wrong ❌
   • C: sign and direction both wrong ❌
   • D: direction wrong ❌

Therefore, the true statement is Option A.