The Dot Product & Projections

Exercises

  1. For the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle,\) calculate \(\bm{u}\cdot\bm{v}.\)
  2. For the vectors \(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle,\) calculate \(\bm{u}\cdot\bm{v}.\)
  3. What is the angle between the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle?\) What is the area of the triangle framed by those vectors?
  4. What is the angle between the vectors \(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle?\)
  5. What is the area of the triangle in the plane with vertices located at the coordinates \((1,1)\) and \((3,-3)\) and \((2,7)?\)
  6. Suppose that \(\bm{u} = \bigl\langle 3,-5 \bigr\rangle\) and that \(\bm{v}\) is some unknown vector. Is it possible that \(\operatorname{proj}_{\bm{u}}(\bm{v}) = \bigl\langle 2,-3\bigr\rangle?\) If it is, determine what \(\bm{v}\) is. If not, why not?
  7. Make a sketch of the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle.\) Then on your sketch add in vectors for the projection of \(\bm{u}\) onto \(\bm{v}\) (\(\operatorname{proj}_{\bm{v}}(\bm{u})\)) and the projection of \(\bm{v}\) onto \(\bm{u}\) (\(\operatorname{proj}_{\bm{u}}(\bm{v})\)), and calculate the components of these vectors explicitly.
  8. Make a sketch of the vectors \(\bm{u} = \langle 5,4 \rangle\) and \(\bm{v} = \langle 1,-3 \rangle.\) Then on your sketch add in vectors for the projection of \(\bm{u}\) onto \(\bm{v}\) (\(\operatorname{proj}_{\bm{v}}(\bm{u})\)) and the projection of \(\bm{v}\) onto \(\bm{u}\) (\(\operatorname{proj}_{\bm{u}}(\bm{v})\)), and calculate the components of these vectors explicitly.

Problems & Challenges

  1. If \(\bm{v} = \langle 3,2 \rangle\) and \(\bm{u}\cdot\bm{v} = 5\) and the angle between \(\bm{u}\) and \(\bm{v}\) is 30°, what must \(|\bm{u}|\) be? Explicitly, in terms of its components, what vector might \(\bm{u}\) be?
  2. Suppose that \(\bm{u} = \bigl\langle 5,7 \bigr\rangle\) and that the \(\bigl|\operatorname{proj}_{\bm{u}}(\bm{v})\bigr| = 3.\) Write down three different possibilities, expressed explicitly in terms of its components, for the vector \(\bm{v}.\)