Page 301 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(c) Make a sketch showing x and in the case where T is the orthogonal projection on the x-axis.

Derive the standard matrices for the rotations about the x-axis, y-axis, and z-axis in from Formula 17.
23.

Use Formula 17 to find the standard matrix for a rotation of radians about the axis determined by the vector
24. .

Note Formula 17 requires that the vector defining the axis of rotation have length 1.

Verify Formula 21 for the given linear transformations.
25.

(a) and

(b) and

(c) and

It can be proved that if A is a matrix with and such that the column vectors of A are orthogonal and have

26. length 1, then multiplication by A is a rotation through some angle . Verify that

satisfies the stated conditions and find the angle of rotation.

The result stated in Exercise 26 is also true in : It can be proved that if A is a matrix with and such that

27. the column vectors of A are pairwise orthogonal and have length 1, then multiplication by A is a rotation about some axis of

rotation through some angle . Use Formula 17 to show that if A satisfies the stated conditions, then the angle of rotation

satisfies the equation

Let A be a matrix (other than the identity matrix) satisfying the conditions stated in Exercise 27. It can be shown tha

28. if x is any nonzero vector in , then the vector determines an axis of rotation when u is

positioned with its initial point at the origin. [See “The Axis of Rotation: Analysis, Algebra, Geometry,” by Dan Kalman,
Mathematics Magazine, Vol. 62, No. 4, October 1989.]

(a) Show that multiplication by