Video: Finding the Matrix of Linear Transformation of Rotating Vectors in Two Dimensions through a Given Angle

Find the matrix for the linear transformation that rotates every vector in ℝ² through an angle of πœ‹/3.

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Video Transcript

Find the matrix for the linear transformation that rotates every vector in ℝ squared through an angle of πœ‹ over three.

This kind of rotation takes each vector and moves them to a new position by rotating them about a certain angle. We can use a rotation matrix here. We can rotate the vectors ℝ by πœƒ using this matrix, cos πœƒ, sin πœƒ, negative sin πœƒ, cos πœƒ.

If we want to rotate about πœ‹ over three, we’re making a 60-degree rotation. We would take cos of πœ‹ over three, sin of πœ‹ over three, negative sin of πœ‹ over three, and then cos of πœ‹ over three again. We know cos of 60 degrees equals one-half. And sin of 60 degrees equals the square root of three over two. Using that information, we plug in what we know.

And we’ll have one-half, square root of three over two, negative square root of three over two, and one-half. This is the matrix we would need to rotate every vector by an angle of πœ‹ over three.

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