Math  /  Geometry

QuestionFind the matrix AA of the rotation about the xx-axis through an angle of π2\frac{\pi}{2}, counterclockwise as viewed from the positive xx-axis.

Studdy Solution

STEP 1

What is this asking? We need to find a special matrix that rotates points in 3D space around the x-axis by a quarter turn. Watch out! Make sure we rotate in the correct direction and remember right-hand rule!

STEP 2

1. Recall the rotation matrix formula.
2. Plug in the given angle.
3. Calculate the matrix elements.

STEP 3

Alright, let's **start** by remembering the general formula for a rotation matrix about the x-axis by an angle θ\theta.
This magical matrix transforms any point by rotating it around the x-axis by θ\theta.
It's like spinning a top, but with math!

STEP 4

The **rotation matrix** Rx(θ)R_x(\theta) is given by: Rx(θ)=[1000cosθsinθ0sinθcosθ]R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{bmatrix}

STEP 5

Now, we're given that our rotation angle θ\theta is π2\frac{\pi}{2}.
So, let's **substitute** that into our general formula.
This is like setting the speed of our spinning top!

STEP 6

Substituting θ=π2\theta = \frac{\pi}{2} into Rx(θ)R_x(\theta), we get: [ R_x\left(\frac{\pi}{2}\right) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\left(\frac{\pi}{2}\right) & -\sin\left(\frac{\pi}{2}\

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