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Exercises 1.5 Exercises

1.

  1. Show that \(\norm{\vu-\vv}^2 = \norm{\vu}^2 + \norm{\vv}^2 - 2\vu \cdot \vv.\)
  2. Deduce the law of cosine from the equation above. This shows that our definition of angle between vectors agrees with the usual one from plane geometry.
A triangle with sides labeled as vectors u, v, and u minus v, with angle theta between u.
Figure 1.5.1. Hint for deducing the Law of Cosines.

2.

Let \(\vu = \gv{1,-2,3}\) and \(\vv = \gv{4,0,-1}\text{.}\)
  1. Compute \(2\vu-\vv\text{.}\)
  2. Compute \(\norm{\vu}\) and \(\norm{\vv}\text{.}\)
  3. Find a unit vector in the direction of \(\vu\text{.}\)

3.

Let \(\vu = \gv{2,1,-1}\) and \(\vv = \gv{1,-1,2}\text{.}\)
  1. Compute \(\vu\cdot\vv\text{.}\)
  2. Find the angle between \(\vu\) and \(\vv\text{.}\)
  3. Find the projection of \(\vu\) onto \(\vv\text{.}\)
  4. Find the component of \(\vu\) orthogonal to \(\vv\text{.}\)

4.

Find an equation of the plane through the point \((2,-1,3)\) with normal vector \(\gv{1,4,-2}\text{.}\) Write your answer both in the form \(\vn\cdot(\vr-\vr_0)=0\) and in the coordinate form \(ax+by+cz=d\text{.}\)

5.

Consider the vectors \(\vu=\gv{1,2,1}\) and \(\vv=\gv{2,-1,3}\text{.}\)
  1. Compute \(\vu\times\vv\text{.}\)
  2. Find the area of the parallelogram spanned by \(\vu\) and \(\vv\text{.}\)
  3. Find an equation of the plane through the point \((1,0,2)\) parallel to both \(\vu\) and \(\vv\text{.}\)

6.

Find an equation of the plane through the three points \(A=(1,2,0)\text{,}\) \(B=(3,1,1)\text{,}\) and \(C=(0,1,2)\text{.}\) Then compute the area of triangle \(ABC\text{.}\)

7.

Let \(\vu = \gv{1,0,2}\text{,}\) \(\vv = \gv{0,1,1}\text{,}\) and \(\vw = \gv{2,1,0}\text{.}\)
  1. Compute the scalar triple product \((\vu\times\vv)\cdot\vw\text{.}\)
  2. Find the volume of the parallelepiped determined by \(\vu,\vv,\vw\text{.}\)
  3. Decide whether the ordered triple \(\vu,\vv,\vw\) is right-handed or left-handed.

8.

  1. Find a parameterization of the line through the points \((1,-1,2)\) and \((4,1,0)\text{.}\)
  2. Find a parameterization of the plane through \((0,1,2)\) parallel to the vectors \(\gv{1,1,0}\) and \(\gv{2,0,1}\text{.}\)
  3. Find a parameterization of the cylinder \(x^2+y^2=9\) for \(-1 \le z \le 2\text{.}\)

9.

  1. Give a parameterization of the sphere \(x^2+y^2+z^2=9\text{.}\)
  2. Give a parameterization of the cone \(z=\sqrt{x^2+y^2}\) for \(0 \le z \le 3\text{.}\)
  3. Give a parameterization of the torus with major radius \(4\) and minor radius \(2\text{.}\)
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