We are given a line passing through the origin with direction ratios 2, 5, and 3, and we are tasked with determining the angles through which this line must be rotated about the x-axis and y-axis to coincide with the z-axis.

Step 1: Direction ratios of the line

The direction ratios of the line are $\mathbf{d} = (2, 5, 3)$. This means the line is directed along a vector with components $2$, $5$, and $3$.

Step 2: Final direction ratios

We want to rotate the line so that it coincides with the z-axis. The direction ratios for the z-axis are $(0, 0, 1)$.

Step 3: Understanding the rotations

• Rotation about the x-axis: Rotating the line about the x-axis will affect only the y and z components of the direction ratios. The x-component remains unchanged.

• Rotation about the y-axis: Rotating the line about the y-axis will affect the x and z components of the direction ratios. The y-component remains unchanged.

We will need to determine these two angles such that, after applying the rotations, the final direction ratios match the z-axis direction $(0, 0, 1)$.

Step 4: Rotation about the x-axis

When rotating a vector $(x, y, z)$ about the x-axis by an angle $\theta$, the new coordinates $(x', y', z')$ are given by the transformation:

$x' = x, \quad y' = y \cos(\theta) - z \sin(\theta), \quad z' = y \sin(\theta) + z \cos(\theta)$

Here, the initial direction ratios are $(2, 5, 3)$, so:

$x' = 2, \quad y' = 5 \cos(\theta) - 3 \sin(\theta), \quad z' = 5 \sin(\theta) + 3 \cos(\theta)$

For the final direction ratios to coincide with the z-axis, we need $x' = 0$ and $y' = 0$, since the x and y components of the z-axis are zero.

Thus, the conditions are: $x' = 2 = 0 \quad (\text{which is already satisfied})$

$y' = 5 \cos(\theta) - 3 \sin(\theta) = 0$ This equation simplifies to: $5 \cos(\theta) = 3 \sin(\theta)$ Dividing both sides by $\cos(\theta)$, we get: $5 = 3 \tan(\theta)$ So: $\tan(\theta) = \frac{5}{3}$

Thus, the angle $\theta$ is: $\theta = \tan^{-1}\left(\frac{5}{3}\right)$ Using a calculator: $\theta \approx 59.04^\circ$

Step 5: Rotation about the y-axis

Now, rotating about the y-axis will affect the x and z components of the vector, and we need to ensure that after this rotation, the vector lies along the z-axis. We already know the vector after the first rotation has the form $(0, 0, z')$, where $z'$ is the final z-component.

The rotation about the y-axis is given by: $x' = x \cos(\phi) + z \sin(\phi), \quad y' = y, \quad z' = -x \sin(\phi) + z \cos(\phi)$

Substituting the values $(x, y, z) = (0, 5, 3)$ and ensuring that the result lies along the z-axis (so $x' = 0$ and $y' = 0$):

The equation for $x' = 0$ becomes: $2 \cos(\phi) + 3 \sin(\phi) = 0$

which simplifies to: [ \cos(\phi) = - sin (\phi