Write down the 3 ×3 matrix Xψrepresenting a clockwise rotation in ℝ^3 around the x-axis by angle

Write down the 3 ×3 matrix Xψrepresenting a clockwise...

Key Concepts

Rotation Matrix in Three Dimensions

A rotation matrix in three dimensions is a special kind of transformation matrix that, when multiplied by a vector, rotates it about a specified axis. These matrices are orthogonal with a determinant of 1, which ensures length and angle are preserved during the rotation. They provide a compact and efficient way to represent rotations in R^3 and are widely used in applications such as computer graphics, robotics, and aerospace engineering.

Rotation About a Coordinate Axis

A rotation about a coordinate axis involves turning points or vectors around one of the main axes (x, y, or z) in R^3. This simplifies the rotation matrix because all the complex interactions occur in the plane perpendicular to the axis of rotation, leaving the coordinate along the axis unchanged. For example, a rotation around the x-axis affects only the y and z components, while the x components remain constant.

Direction of Rotation (Clockwise vs. Counterclockwise)

The sense of rotation, whether clockwise or counterclockwise, determines the sign of the angle used in the rotation matrix. Conventionally, in right-handed coordinate systems, positive angles correspond to counterclockwise rotations while negative angles signify clockwise rotations. Specifying a clockwise rotation requires adjusting the angle's sign appropriately to ensure the resulting matrix rotates vectors in the intended direction.

Transcript

0:00 Hello there.

00:01 So in this exercise we need to take the general rotation about an axis defined by some unitary vector u and an angle of theta and use this rotation to reduce to the usual rotations about the corresponding axis so for example here the first thing that we need to do is obtain the rotation about the x -axis of 90 degrees so let's work out this if our unitary vector is u equals to one zero zero that means that a is equal to 1 b is equal to c and both are equals to 0 and the fact that we have here theta equals 90 degrees means that the cosign of 90 degrees is equals to 0 and the sign of 90 degrees is equals to 1 so having this information we can start to replace the values on this general rotation matrix so then here we're going to call this the rotation about the x -axis of 90 degrees so here we have the data all the cosines will be equals to zero so we're going to eliminate here these cosines here is the same we will have here just one here as well just one you're going to to delete all the cosines because all of them are equals to zero.

01:39 And we have a more easy expression.

01:44 The cosines are eliminated here.

01:50 And these signs will take the value of one.

01:53 So it's multiplying some coefficients here, b and c, so we can erase them as well because they will take the value of one.

02:02 So it's not relevant in this matrix.

02:06 Then let's take the values for b and c equals to zero so that means that this value will be zero b this will be zero this as well this as well b zero zero zero be zero zero so you can see that we end with just some few numbers.

02:35 So here we have a square, this is equal to 1, square.

02:40 Here we have minus a, so minus 1, and here plus a.

02:48 And the remaining values are 0.

02:51 So you can see that we can recover the usual matrix rotation about the x -axis of 90 degrees that corresponds to the matrix one zero zero zero minus one zero zero zero zero okay now let's work out for the y axis so the b part corresponds to taking u equals to zero one zero again theta is equals to 90 degrees and that implies that the cosine of of theta will be zero and the sign of theta will be equals to 1...

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