Exercise3

Shown below in FigureĀ 18 are the vectors \(\evec_1\text{,}\) \(\evec_2\text{,}\) and \(\evec_3\) in \(\real^3\text{.}\)

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Figure2.6.18The vectors \(\evec_1\text{,}\) \(\evec_2\text{,}\) and \(\evec_3\) in \(\real^3\text{.}\)

  1. Imagine that the thumb of your right hand points in the direction of \(\evec_1\text{.}\) A positive rotation about the \(x\) axis corresponds to a rotation in the direction in which your fingers point. Find the matrix definining the matrix transformation \(T\) that rotates vectors by \(90^\circ\) around the \(x\)-axis.

  2. In the same way, find the matrix that rotates vectors by \(90^\circ\) around the \(y\)-axis.

  3. Find the matrix that rotates vectors by \(90^\circ\) around the \(z\)-axis.
  4. What is the cumulative effect of rotating by \(90^\circ\) about the \(x\)-axis, followed by a \(90^\circ\) rotation about the \(y\)-axis, followed by a \(-90^\circ\) rotation about the \(x\)-axis.

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