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Give the $4 \times 4$ matrix that rotates points in $\mathbb{R}^{3}$ about the $x$ -axis through an angle of $60^{\circ} .$ (See the figure.)

$\left[\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {0} & {\frac{1}{2}} & {-\frac{\sqrt{3}}{2}} & {0} \\ {0} & {\frac{\sqrt{3}}{2}} & {\frac{1}{2}} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right]$

Algebra

Chapter 2

Matrix Algebra

Section 7

Applications to Computer Graphics

Introduction to Matrices

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

06:52

Give the $4 \times 4$ matr…

02:27

Write the vertex matrix an…

01:03

The coordinates of a point…

01:36

Rotation of Axes Determine…

02:16

Determine the angle of rot…

01:26

Rotate the figure with the…

05:53

Determine the $X Y$ -coord…

01:13

02:35

01:29

So this question we first constructs the three by three matrix for the rotation. No, from the DAG girl that is looking at the door. A groom by the DAG room. We have duds defect. So do you want these roots instead? 30 degrees. So would I. The negative why exists on it stops odds stopping odds. The points, um co sign off minus 30 degrees. Sign off minus 30 degrees on zero. This is equality births three over itsu minus one over TSU on an zero similarly defect. So it to he's roots. It said Discredited 60 degrees because it's going toward the positive X exists. If you look at the diagram, stop on it stops arts stopping ads at the points course sign of 60 degrees sign of 60 degrees, another zero. And then this is just 1/4 1 of its suit words. Three over to sue under zero and, lastly, defect. So eat three e three. He's not changed by the rotation is not changed by the rotation because it is already on the Z exists on day recalled on the question Is accident goes to rotate about the exits, so since it the reserve that is the exists. The Vector eatery is not changed by the rotation. This implies that if we bring all this together, the Matrix for rotation is given by a, which is a quote. So the 1st 1 is what's three over to sue, minus one over to sue and zero. Then eat to give us one more word. Sue for three over two and zero and that Peter is unchanged. So this implies that's in Virginia's coordinates. In more Gina's coordinates. Dirac position is represented by by the mid tricks. A zero transpose zero wall, which is just This is my ace on average. Three over zoo one over TSU zero minus one over to sue Rude story over tsu 00 zero. Why are so? This is my age and I are zero 00 on then zero is there? Is there anyone? No for lying. Example. Seven B. In this section, we see that in a more Gina's coordinates. In a more genius coordinates The translation Victor. The translation of the translation by Vessel five minus two Comma Juan. He's represented Bi Metrics Juan 00001000010 And they add five minus 211 So, with this we know have that's this implies That's the complete transformation. Easy represented in Omagh. Gina's coordinates Ward needs by the metrics never moved out of the next page. So we're gonna multiply the translation, which is 1000 0100 00105 minus 211 on. Then we multiplied Thies. Bye, Woods. Theory of it. Sue minus one over. TSU 00 won over soon with three over Sue. 00 00100001 When you're multiplied this you're gonna have words three over the zoo minus one over TSU 00 won over TSU with three over 200 0010 five minus to want one on. This is the answer.

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