Exercise1

Shown in Figure 11 are two vectors \(\vvec_1\) and \(\vvec_2\) in the plane \(\real^2\text{.}\)

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Figure3.2.11Vectors \(\vvec_1\) and \(\vvec_2\) in \(\real^2\text{.}\)
  1. Explain why \(\bcal=\{\vvec_1,\vvec_2\}\) is a basis for \(\real^2\text{.}\)

  2. Using Figure 11, indicate the vectors \(\xvec\) such that

    1. \(\coords{\xvec}{\bcal} = \twovec{2}{-1}\)

    2. \(\coords{\xvec}{\bcal} = \twovec{-1}{-2}\)

    3. \(\coords{\xvec}{\bcal} = \twovec{0}{3}\)

  3. Using Figure 11, find the representation \(\coords{\xvec}{\bcal}\) if

    1. \(\xvec = \twovec{-2}{-1}\text{.}\)

    2. \(\xvec = \twovec{2}{4}\text{.}\)

    3. \(\xvec = \twovec{2}{-5}\text{.}\)

  4. Find \(\coords{\xvec}{\bcal}\) if \(\xvec=\twovec{60}{90}\text{.}\)

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