Use matrix multiplication to find the image of the vector...

Use matrix multiplication to find the image of the vector (2,-1,2) if it is rotated (a) $30^{\circ}$ counterclockwise about the positive $x$ -axis. (b) $30^{\circ}$ clockwise about the positive $y$ -axis. (c) $45^{\circ}$ counterclockwise about the positive $y$ -axis. (d) $90^{\circ}$ clockwise about the positive $z$ -axis.

Use matrix multiplication to find the image of the vector (2,-1,2) if it is rotated
(a) $30^{\circ}$ counterclockwise about the positive $x$ -axis.
(b) $30^{\circ}$ clockwise about the positive $y$ -axis.
(c) $45^{\circ}$ counterclockwise about the positive $y$ -axis.
(d) $90^{\circ}$ clockwise about the positive $z$ -axis.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where a matrix (representing a linear transformation) is multiplied by a vector to produce another vector. It is the key tool for applying transformations such as rotations to vectors by combining the effects of the transformation with the original vector’s components.

Rotation Matrices

Rotation matrices are specific matrices used to perform rotations in a coordinate space. In three-dimensional space, distinct rotation matrices are formulated for rotations about the x, y, or z axes. They incorporate trigonometric functions that represent the sine and cosine of the rotation angle, ensuring that the transformed vector maintains its magnitude.

Axis of Rotation

The axis of rotation is the fixed line in space about which the object or vector rotates. In three-dimensional transformations, specifying the correct axis of rotation (such as the positive x, y, or z axis) is crucial, because the structure of the rotation matrix directly depends on which axis is used.

Direction of Rotation (Clockwise vs. Counterclockwise)

The direction of rotation, whether clockwise or counterclockwise, determines the sign conventions of the sine terms in the rotation matrices. A clockwise rotation typically inverts the sign compared to a counterclockwise rotation, which is essential to correctly modeling the rotation according to the problem's specifications.

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