Determining matrix which represents Linear Transformation

Question

Let F : E 3 → E 3 be one of the two possible orthogonal rotations of angle 5π/6 around the line of equation 2x + y + 5z = 0 = 4x − y + 10z. I.Determine the matrix representing F II.Determine if such a rotation is a direct isometry. I am struggling to see relation between ''orthogonal rotation'' and linear transformation as i have never encountered a question like this.And i am not sure what is meant by "direct isometry"

Answer 1

HINT

We can proceed as follow

• find the normalized direction vector $\vec v$ for the given line
• complete a basis by two orthonormal vectors $\vec u$ and $\vec w$ orthogonal to $\vec v$
• in the basis $\{\vec u,\vec w,\vec v\}$ the rotation matrix is given by

$$R=\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0 &0 &1\end{bmatrix}$$

• then change basis from $\{\vec u,\vec w,\vec v\}$ to the canonical

Recall that if the orientation stays the same we say that the isometry is direct, if the orientation changes we say that the isometry is opposite.