To solve this problem using the same method described in the example with direction ratios $(1, 1, 1)$, we will follow similar steps for the line with direction ratios $(2, 5, 3)$, which passes through the origin. Let's break down the solution step-by-step:

Step 1: Direction Cosines of the Line

The direction ratios of the line are $(2, 5, 3)$, so we first calculate the direction cosines $(c_x, c_y, c_z)$. These are given by the normalized values of the direction ratios, where: $c_x = \frac{2}{\sqrt{2^2 + 5^2 + 3^2}}, \quad c_y = \frac{5}{\sqrt{2^2 + 5^2 + 3^2}}, \quad c_z = \frac{3}{\sqrt{2^2 + 5^2 + 3^2}}$

First, calculate the magnitude of the direction ratios: $\text{Magnitude} = \sqrt{2^2 + 5^2 + 3^2} = \sqrt{4 + 25 + 9} = \sqrt{38}$

Now, the direction cosines are: $c_x = \frac{2}{\sqrt{38}}, \quad c_y = \frac{5}{\sqrt{38}}, \quad c_z = \frac{3}{\sqrt{38}}$

Step 2: Angle $\alpha$ with the x-axis

The angle $\alpha$ with the x-axis is the angle between the line and the x-axis. The cosine of this angle is given by: $\cos \alpha = c_x = \frac{2}{\sqrt{38}}$ Thus, $\alpha = \cos^{-1}\left(\frac{2}{\sqrt{38}}\right)$ This evaluates to: $\alpha \approx \cos^{-1}(0.325) \approx 71.57^\circ$

Step 3: Angle $\beta$ with the y-axis

Similarly, the angle $\beta$ with the y-axis is given by the cosine of the angle as: $\cos \beta = c_y = \frac{5}{\sqrt{38}}$ Thus, $\beta = \cos^{-1}\left(\frac{5}{\sqrt{38}}\right)$ This evaluates to: $\beta \approx \cos^{-1}(0.811) \approx 35.10^\circ$

Step 4: Rotation to Coincide with the z-axis

For the line to coincide with the z-axis, it must be rotated in such a way that the line's direction cosines match those of the z-axis (which are $(0, 0, 1)$).

• The rotation about the x-axis will change the angle $\alpha$ to $0^\circ$, which corresponds to aligning the line along the z-axis.
• The rotation about the y-axis will change the angle $\beta$ to $0^\circ$, completing the alignment with the z-axis.

Thus, the line should be rotated:

• Around the x-axis by approximately $71.57^\circ$
• Around the y-axis by approximately $35.10^\circ$

Final Answer:

• Rotation about x-axis: $\alpha \approx 71.57^\circ$
• Rotation about y-axis: $\beta \approx 35.10^\circ$

Would you like any further clarifications on the steps? Let me know if you have any questions!

Related Questions:

1. How would the solution change if the line had direction ratios $(4, 3, 2)$?
2. What happens if the direction ratios are negative, e.g., $(-2, 5, 3)$?
3. How do you interpret the angles between two vectors in 3D space?
4. What is the role of the magnitude of direction ratios in finding the direction cosines?
5. Can this method be generalized to determine angles for any vector in 3D space?

Tip:

In problems involving rotation of vectors, it's useful to visualize the rotations in terms of the coordinate axes and how the direction cosines change as the vector moves. This makes it easier to understand and calculate the required angles.