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MATH / Let v₀ be a vector. The vector v₀ is rotated about the origin by an angle of 42^(∘) counter-clockwise, taking it to vector v₁. The vector v₁ is then reflected over the line with di…

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problem

Let v0\mathbf{v}_0 be a vector. The vector v0\mathbf{v}_0 is rotated about the origin by an angle of 42∘42^\circ counter-clockwise, taking it to vector v1.\mathbf{v}_1. The vector v1\mathbf{v}_1 is then reflected over the line with direction vector $\begin{pmatrix} \cos 108^\circ \\ \sin 108^\circ \end{pmatrix},$ taking it to vector v2.\mathbf{v}_2. The vector v2\mathbf{v}_2 can also be produced by reflecting the vector v0\mathbf{v}_0 over the line with direction vector $\begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix},$ where θ\theta is an acute angle. Find θ.\theta.
Plain-text mathematical notation (without MathML)
Let v₀ be a vector.  The vector v₀ is rotated about the origin by an angle of 42^(∘) counter-clockwise, taking it to vector v₁.  The vector v₁ is then reflected over the line with direction vector $\begin{pmatrix} \cos 108^\circ \\ \sin 108^\circ \end{pmatrix},$ taking it to vector v₂.

The vector v₂ can also be produced by reflecting the vector v₀ over the line with direction vector $\begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix},$ where θ is an acute angle.  Find θ.
Original LaTeX notation
Let $\mathbf{v}_0$ be a vector.  The vector $\mathbf{v}_0$ is rotated about the origin by an angle of $42^\circ$ counter-clockwise, taking it to vector $\mathbf{v}_1.$  The vector $\mathbf{v}_1$ is then reflected over the line with direction vector $\begin{pmatrix} \cos 108^\circ \\ \sin 108^\circ \end{pmatrix},$ taking it to vector $\mathbf{v}_2.$

The vector $\mathbf{v}_2$ can also be produced by reflecting the vector $\mathbf{v}_0$ over the line with direction vector $\begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix},$ where $\theta$ is an acute angle.  Find $\theta.$

level

Level 4

type

Precalculus

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Source and history

task_id
f2287d73-820c-5097-bde6-98d5e999333c
task_key
precalculus--test--f2287d73-820c-5097-bde6-98d5e999333c
task_revision_id
1
config / split
precalculus / test

Official source

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