Math Problem Statement

Consider the line with direction ratios 2,5,3 and passing through origin. Determine the angles through which the line should be rotated about x-axis and then about y-axis so that it coincides with z-axis. perform this question by this method ->Consider the line with direction ratios 1, 1, 1 and passing through the origin. Determine angles through which the line should be rotated about x-axis and then about y-axis; so that it coincides with z-axis. Solution : Direction cosines of the line are (cx, cy, cz) =       1 3, 1 3, 1 3 d2 = c2 y + c2 z ⇒ d = c2 y + c2 z ⇒ d = 1 3 + 1 3 = 2 3 Let the line makes an angle α with x-axis. ⇒ cos α = cz d = 1 2 and sin α = cy d = 1 2 since cos α = sin α = 1 2 ⇒ α = 45° Let the line makes an angle β with y-axis. ⇒ cos β = d = 2 3 β = cos–1 2 3 = 35.26° Thus, α = 45° and β = 35.26°

Solution

To solve this problem using the same method described in the example with direction ratios (1,1,1)(1, 1, 1), we will follow similar steps for the line with direction ratios (2,5,3)(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)(2, 5, 3), so we first calculate the direction cosines (cx,cy,cz)(c_x, c_y, c_z). These are given by the normalized values of the direction ratios, where: cx=222+52+32,cy=522+52+32,cz=322+52+32c_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: Magnitude=22+52+32=4+25+9=38\text{Magnitude} = \sqrt{2^2 + 5^2 + 3^2} = \sqrt{4 + 25 + 9} = \sqrt{38}

Now, the direction cosines are: cx=238,cy=538,cz=338c_x = \frac{2}{\sqrt{38}}, \quad c_y = \frac{5}{\sqrt{38}}, \quad c_z = \frac{3}{\sqrt{38}}

Step 2: Angle α\alpha with the x-axis

The angle α\alpha with the x-axis is the angle between the line and the x-axis. The cosine of this angle is given by: cosα=cx=238\cos \alpha = c_x = \frac{2}{\sqrt{38}} Thus, α=cos1(238)\alpha = \cos^{-1}\left(\frac{2}{\sqrt{38}}\right) This evaluates to: αcos1(0.325)71.57\alpha \approx \cos^{-1}(0.325) \approx 71.57^\circ

Step 3: Angle β\beta with the y-axis

Similarly, the angle β\beta with the y-axis is given by the cosine of the angle as: cosβ=cy=538\cos \beta = c_y = \frac{5}{\sqrt{38}} Thus, β=cos1(538)\beta = \cos^{-1}\left(\frac{5}{\sqrt{38}}\right) This evaluates to: βcos1(0.811)35.10\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)(0, 0, 1)).

  • The rotation about the x-axis will change the angle α\alpha to 00^\circ, which corresponds to aligning the line along the z-axis.
  • The rotation about the y-axis will change the angle β\beta to 00^\circ, completing the alignment with the z-axis.

Thus, the line should be rotated:

  • Around the x-axis by approximately 71.5771.57^\circ
  • Around the y-axis by approximately 35.1035.10^\circ

Final Answer:

  • Rotation about x-axis: α71.57\alpha \approx 71.57^\circ
  • Rotation about y-axis: β35.10\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)(4, 3, 2)?
  2. What happens if the direction ratios are negative, e.g., (2,5,3)(-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.

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Math Problem Analysis

Mathematical Concepts

3D Geometry
Direction Ratios
Direction Cosines
Rotation of Vectors

Formulas

Direction Cosines: cx = a / √(a² + b² + c²), cy = b / √(a² + b² + c²), cz = c / √(a² + b² + c²)
Cosine of angle with axis: cos(α) = cx, cos(β) = cy
Rotation formulas for 3D space

Theorems

Direction Cosine Theorem
Rotation of Vectors in 3D

Suitable Grade Level

Grades 10-12