Rotations

and can be written in column vector form as

and

Now consider a specific rotation:

counterclockwise.

So our matrix equation looks like this

Solving this we find that

We also wish to rotate the unit basis vector so that it becomes the new unit vector

Solving the equation

we find that

Thus, our transformation matrix

must be

Now consider any vector

To transform this general vector we can also use the same

because

so

The transformation matrix has a particular form based on the cosines of the angles between the new axes and the old axes. Specifically,

the cosine of the angle between and

For this reason, the matrix

and is the same as the negative of the sine of the rotation angle. So, what this means is that we can get a simpler form of the 2-dimensional transformation matrix. If we wish to rotate our basis vectors by an angle of

then we can use the transformation matrix

For example, to rotate our basis vectors

we would use the rotation matrix

It is important that a rotation matrix does not stretch or shrink the points; it only rotates them. We are guaranteed that this will happen whenever

and you can see that this will always be the case when we use

as defined above.

plane. So we need a transformation matrix such that

So our transformation equations are

and

and in terms of and . In other words, we need

We can easily calculate that

So our appropriate transformation equations become

and

So we substitute our transformation equations into the parabola equation to get

This equation is graphed at right.

STEP 2013, Math I, #5: The point P has coordinates which satisfy

(i) Sketch the locus of P in the case

, giving the points of intersection with the coordinate axes.

, let Q be the point obtained by rotating P clockwise about the origin by an angle , so that the coordinates of Q are given by

, giving the equation of the line of symmetry.

Skill Set

Rotations

No. 1

Rotations in 3-space

For rotations around the coordinate axes in 3-space we can adapt our 2-dimensional rotation matrix. Rotations in space have to specify a rotation axis. We will define rotations about the coordinate axes.

Imagine yourself looking down at the xy-plane from the viewpoint of the positive z-axis. Then a rotation counterclockwise around this axis is equivalent to our previous counterclockwise rotation in the xy-plane. We can use our previous matrix only expanded to accommodate three components.

Imagine now you are looking at the yz-plane from the viewpoint of the positive x-axis. We can modify our previous matrix to rotating around the x-axis, thereby leaving the x-coordinates of all points/vectors unchanged.

Finally, if you are looking at the zx-plane from the viewpoint of the positive y-axis and you wish to rotate around the y-axis an angle of

you use

You can also define rotations about other axes. The trick is to figure out, as we did in our first example above, what happens to the unit basis vectors

and parallel to the vectors

, .

These are drawn at left.

You can verify that

so they are orthogonal.

We now need a third vector that is orthogonal to both of these. We use the cross product:

These three mutually orthogonal vectors are shown at left.

Now all we need to do is to make our vectors into unit vectors and we have our new basis vectors

, and

That means that the transformation matrix

will rotate any vector in the same way it rotated the original unit basis vectors. You can verify that this is a valid transformation matrix by checking its determinant. Expansion along the third column produces

, and lie on a sphere of unit radius centred at the origin, , which is fixed. Initially, is at , is at

turns directly towards the positive y-axis through an angle . The position of after this rotation is denoted by

. Write down the coordinates of .

after this rotation is denoted by . Find the coordinates of . Find also the coordinates of the points and , which are the positions of and

after the two rotations.

(iii) The sphere is now rotated for a third time, so that P returns from to its original position . During the rotation , P remains in the plane containing , , and

. Show that the angle of this rotation , , satisfies

and find a vector in the direction of the axis about which this rotation takes place.