Topics

  1. Know how to use vectors to determine if 3 points are collinear and how to find the area of a triangle given 3 points.
  2. Know how to find the component form and magnitude of a vector.
  3. Know how to find the angle between vectors or 2 planes.
  4. Know if vectors are parallel, orthogonal, or neither.
  5. Find the cross product between 2 vectors.
  6. Write the equation of the line between 2 points.
  7. Write the equation of a plane given 3 points, or a point and the normal vector, or 2 points and a parallel/perpendicular vector.
  8. Know what the different parts of the vector form of the equation of a line or plane stand for.
  9. Be able to convert between the different forms of a line and plane.
  10. Be able to draw pictures that represent vectors, lines, planes, ect. in 3d.
  11. Vector operations in 3d.

Examples

  1. Are the given 3 points collinear?
    1. \((6, 3, -1), (5, 8, 3), (7, -2, -5)\)
    2. \((5, 2, 0), (2, 6, 1), (2,4,7)\)
  2. Find the area of the triangle that goes through the points \((2, -1, 1)\), \((5, 1, 4)\), and \((0,1,1)\).
  3. For the points \(A(5, -1, 2)\) and \(B(-3, 3, 0)\):
    1. Plot each point in the three-dimensional coordinate system.
    2. Find the component form of vector \(\vec{AB}\).
    3. Find \(\vert\vert\vec{AB}\vert\vert\), the maginitude of vector \(\vec{AB}\).
  4. Find the angle between
    1. \(\vec{u}=\langle -2, -1, 0 \rangle\) and \(\vec{v}=\langle 1,-2,1 \rangle\).
    2. \(\vec{u}=\langle 3, 1, -1 \rangle\) and \(\vec{v}=\langle 4, 5, 2 \rangle\).
  5. Find the angle of intersection for the following pairs of planes.

    1. \(\begin{align} \phantom{3}x + \phantom{3}y - 2z &= 0\\ 2x - \phantom{3}y + 3z &= 0 \end{align}\)

    2. \(\begin{align} 3x - 4y + 5z &= 6\\ \phantom{3}x + \phantom{3}y - \phantom{3}z &= 2 \end{align}\)
  6. Are the vectors \(\mathbf{u}\) and \(\mathbf{v}\) parallel, orthogonal, or neither?
    1. \(\mathbf{u}=3\mathbf{j}+2\mathbf{k}\) and \(\mathbf{v}=12\mathbf{i}-18\mathbf{k}\)
    2. \(\mathbf{u}=\begin{pmatrix} \phantom{-}39 \\ -12\\ \phantom{-}21 \end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix} -26 \\ \phantom{-1}8\\ -14 \end{pmatrix}\)
    3. \(\mathbf{u}=\langle 6, 5, 9\rangle\) and \(\mathbf{v}=\langle 5, 3, -5\rangle\).
  7. Find the cross product, \(\vec{u} \times \vec{v}\).
    1. \(\mathbf{u}=\langle -2, 8, 2\rangle\) and \(\mathbf{v}=\langle 1, 1, -1\rangle\)
    2. \(\mathbf{u}=2\mathbf{i}+3\mathbf{j}+2\mathbf{k}\) and \(\mathbf{v}=3\mathbf{i}+\mathbf{j}+2\mathbf{k}\)
  8. Find a unit vector orthogonal to both \(\mathbf{u}=\begin{pmatrix}\phantom{-}2\\-1\\ \phantom{-}1\end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix}-1\\\phantom{-}1\\ -2\end{pmatrix}\).
  9. Find the equation of the line that passes through the points \(P(2, 0, 2)\) and \(Q(1, 4, -3)\).
    1. What is the direction vector?
    2. Give the equation in…
      1. vector form
      2. parametric form
      3. symmetric/cartesian form
  10. Find the general form of the equation of the plane passing through the points \((4, -1, 3)\), \((2, 5, 1)\), and \((-1, 2, 1)\).
  11. Find the general form of the equation of the plane passing through point \((-4, 5, 2)\) with normal vector \(\vec{n}=\begin{pmatrix}-1\\\phantom{-}2\\\phantom{-}1\end{pmatrix}\).
  12. Find the equation of the plane through points \(P(1,2,0)\) and \(Q(-1, -1, 2)\), perpendicular to \(2x-3y + z= 6\).
  13. Given \(\mathbf{u}=\langle 6, 5, 9\rangle\) and \(\mathbf{v}=\langle 5, 3, -5\rangle\), find \(2\mathbf{u}-\mathbf{v}\).
  14. Find the volume of the parallelpiped with vertices \(A(0,0,0)\), \(B(2,0,0)\), \(C(2, 4, 0)\), \(D(0,4,0)\), \(E(0,0,6)\), \(F(2,0,6)\), \(G(2,4,6)\), and \(H(0, 4, 6)\).
  15. Find the distance between the point and the plane.
    1. \((1, 3, 4)\) and \(4x-5y+2z=6\).
    2. \((-1,3,-6)\) and \(x - 2y + 2z =3\)

Answers to Examples above

  1.  
    1. yes
    2. no
  2. \(\sqrt{43}\approx6.56\)
    1. See plot at the end of this document.

  1. \(\begin{pmatrix} -8\\ \phantom{-}4\\ -2\\ \end{pmatrix}\)

  2. 9.17
  1.  
    1. \(90^\circ\)
    2. \(47.61^\circ\)
  2.  
    1. \(56.9^\circ\)
    2. \(60.7^\circ\)
  3.  
    1. Neither
    2. Parallel
    3. Orthogonal
  4. \(\vec{u} \times \vec{v}=\)
    1. \(\langle -10, 0, -10\rangle\)
    2. \(4\mathbf{i}+2\mathbf{j}-7\mathbf{k}\)
  5. \(\begin{pmatrix}\frac{\sqrt{11}}{11}\\ \frac{3\sqrt{11}}{11}\\ \frac{\sqrt{11}}{11}\end{pmatrix}\).
  6.  
    1. \(\vec{v}=\langle -1, 4, -5 \rangle\)
    2.  
      1. \(\begin{pmatrix}x\\y\\z\\\end{pmatrix}=\begin{pmatrix}2\\0\\2\\\end{pmatrix}+t \begin{pmatrix}-1\\\phantom{-}4\\-5\\\end{pmatrix}\)
      2. \(\begin{align} \\ x &= 2-t \\ y &= 4t\\z&=2-5y \\ \ & \ \end{align}\)
      3. \(\dfrac{x-2}{-1}=\dfrac{y}{4}=\dfrac{z-2}{-5}\)

  7. \(-x+y + 4z = 7\) or \(-x+y + 4z - 7 = 0\).

  8. \(-x+2y+z=16\)

  9. \(x+2y+4z=5\)

  10. \(2\mathbf{u}-\mathbf{v} = \langle 7, 7, 23 \rangle\).

  11. 48
  12.  
    1. \(\frac{3\sqrt{5}}{5}\)
    2. \(\frac{22}{3}\)

Solutions to the answers above