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Section 5 Wrap Up Questions

  1. Decide whether each of the following statements is true or false. Give a reason for your answer.

    1. If \(\vec{v}\cdot \vec{w}=||\vec{v}|| ||\vec{w}||\text{,}\) then \(\vec{v}\) and \(\vec{w}\) point in the same direction.
    2. If \(\vec{v}\) and \(\vec{w}\) point in opposite directions, then \(\vec{v}\cdot \vec{w}=0\text{.}\)
    3. If \(\vec{v} \cdot \vec{w} =\vec{a} \cdot \vec{b}\text{,}\) then the angle between \(\vec{v}\) and \(\vec{w}\) is the same as the angle between \(\vec{a}\) and \(\vec{b}\text{.}\)
  2. Often when solving trigonometric functions, we need to be careful we get the correct solution in the correct quadrant. Explain why when finding the angle between vectors, we do not need to worry about this issue and can instead directly use the inverse cosine function.
  3. If a dot product of \(\vec{v}\) and \(\vec{w}\) is a large positive number, why would it be invalid to conclude they point in roughly the same direction? What further information would you need to decide if they did?
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