b. Find the standard matrix for the reflection of I R² about the line that makes an angle of ? / 4 ( = 45° ) with the positive x-axis, and then use that matrix to find the reflection of (1, 2) onto this line.
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The angle between $L$ and the positive x-axis is $\theta = \frac{\pi}{4}$. The standard matrix for the reflection about a line that makes an angle $\theta$ with the positive x-axis is given by: $R_\theta = \begin{bmatrix} \cos(2\theta) & \sin(2\theta) \\ Show more…
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Find the standard matrix for the reflection of $R^{2}$ about the stated line, and then use that matrix to find the reflection of the given point about that line. The reflection of (1,2) about the line that makes an angle of $\pi / 4\left(=45^{\circ}\right)$ with the positive $x$ -axis.
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Problem 3. Find the matrix of the reflection across the plane x + 2y + 3z = 0 in standard coordinates. Some steps: (1) The plane is the kernel of the 1 x 3 matrix (1 2 3). Find a basis for this kernel. You can also do this by simply solving x + 2y + 3z = 0 as a system of one equation with 3 unknowns. (2) Let v1, v2 be the basis you found above, and add v3 = (1 2 3)^T. This vector is perpendicular (in 3D space) to v1 and to v2 (in fact it is in the normal direction of the plane x + 2y + 3z = 0). Let B = (v1, v2, v3). This is a basis of R^3. Write the matrix B of the reflection with respect to the coordinates B (or B, B is you prefer). (3) Let T be the matrix with columns v1, v2, v3. Is the answer to the question T^-1BT, or is it TBT^-1?
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