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AI HL · AHL 4

Vectors

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AHL 4.18

Parallel vectors, perpendicular vectors and area

AHL 4.18

Parallel vectors, perpendicular vectors and area

Target: I can say what \(\mathbf{a}\times\mathbf{b}\) is for parallel vectors and for perpendicular vectors. (+4 more)

By the end of this section

Key points

Parallel and perpendicular vectors

  • If \(\mathbf{a}\) and \(\mathbf{b}\) are parallel, \(\mathbf{a}\times\mathbf{b}=\mathbf{0}\). In particular \(\mathbf{a}\times\mathbf{a}=\mathbf{0}\). It is the zero vector, not the number 0. (The vector product is also called the cross product.)
  • If \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular, \(|\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\), because \(\sin90^\circ=1\).
  • So a vector product of \(\mathbf{0}\) tells you two vectors are parallel. For three points, \(\overrightarrow{PQ}\times\overrightarrow{PR}=\mathbf{0}\) means P, Q and R lie on one line.

Area

  • \(|\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta\) is the area of the parallelogram with sides \(\mathbf{a}\) and \(\mathbf{b}\).
  • A parallelogram splits into two equal triangles, so the triangle with sides \(\mathbf{a}\) and \(\mathbf{b}\) has area \(\tfrac12|\mathbf{a}\times\mathbf{b}|\). This is \(\tfrac12ab\sin C\) written with vectors.
  • Use it when you have the vertices in 3D and no angle. You do not need the angle.

Method for a triangle from its vertices

  • Sketch the triangle. Pick one vertex and write the two sides from it as vectors. It does not matter which two sides you use.
  • Find the vector product of the two sides, then its magnitude, then halve it.
  • Common errors: forgetting the half, and getting the middle component of the vector product wrong.
  • Check: your vector product must have scalar product 0 with both sides.

Explore: the vector product as an area

The vectors a and b lie in the plane z = 0, the page. Drag their tips. a × b is perpendicular to both, so it points straight out of the page or into it: the orange symbol shows which, and the edge-on view below the grid shows it as an arrow whose length is the area of the parallelogram. Before you swap a and b, say what will happen to the arrow.

a × b
Parallelogram (both shaded)
Triangle (the gold part)

⊙ means a × b points out of the page, towards you (+k). ⊗ means into the page (−k). Right-hand screw rule: turn a screw from a towards b (the orange arrow) and it moves the way a × b points. Click a tip, then use the arrow keys to move it one unit.

Drag a tip to change a vector.

  1. Swap a and b. What happens to a × b, to the arrow in the edge-on view and to the area? Why?
  2. Make b turn from anticlockwise of a to clockwise of a. What happens to the sign of the k part, and why does the area stay positive?
  3. Make a and b parallel. What happens to the parallelogram, to a × b and to sin θ?
  4. Make a and b perpendicular. What shape is the parallelogram, and what is |a × b| in terms of |a| and |b|?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Area of a triangle from its vertices

Find the area of the triangle with vertices \(P(2, 1, 0)\), \(Q(4, 3, -1)\), \(R(1, 5, 3)\).

PQR
Working
  1. Two sides as vectors

    \(\mathbf{a}=\overrightarrow{PQ}=\begin{pmatrix}2\\2\\-1\end{pmatrix}\), \(\mathbf{b}=\overrightarrow{PR}=\begin{pmatrix}-1\\4\\3\end{pmatrix}\)

    Any two sides from one vertex will do. There is no angle given, so you cannot use \(\tfrac12ab\sin C\).

  2. The vector product

    \(\mathbf{a}\times\mathbf{b}=\begin{pmatrix}2\\2\\-1\end{pmatrix}\times\begin{pmatrix}-1\\4\\3\end{pmatrix}=\begin{pmatrix}2\times3-(-1)\times4\\ (-1)\times(-1)-2\times3\\ 2\times4-2\times(-1)\end{pmatrix}=\begin{pmatrix}10\\-5\\10\end{pmatrix}\)

    Work the components out carefully. The middle one runs the other way round.

  3. Its magnitude

    \(|\mathbf{a}\times\mathbf{b}|=\sqrt{225}=15\)

    This is the area of the parallelogram on a and b.

  4. Half of it

    Area \(=\tfrac12|\mathbf{a}\times\mathbf{b}|=7.50\) square units

    The triangle is half the parallelogram. 3 significant figures.

Video walkthrough coming
Area of a parallelogram

ABCD is a parallelogram with \(A(1, 0, 2)\), \(B(4, 2, 1)\) and \(C(5, 5, 4)\). Find the coordinates of D, then the area of the parallelogram.

ABCD
Working
  1. The fourth vertex

    \(\overrightarrow{AD}=\overrightarrow{BC}=\begin{pmatrix}1\\3\\3\end{pmatrix}\), so \(D=(2, 3, 5)\)

    Opposite sides of a parallelogram are equal vectors.

  2. Two sides from A

    \(\mathbf{a}=\overrightarrow{AB}=\begin{pmatrix}3\\2\\-1\end{pmatrix}\), \(\mathbf{b}=\overrightarrow{AD}=\begin{pmatrix}1\\3\\3\end{pmatrix}\)

    Use the two sides that meet at A.

  3. The vector product

    \(\mathbf{a}\times\mathbf{b}=\begin{pmatrix}3\\2\\-1\end{pmatrix}\times\begin{pmatrix}1\\3\\3\end{pmatrix}=\begin{pmatrix}2\times3-(-1)\times3\\ (-1)\times1-3\times3\\ 3\times3-2\times1\end{pmatrix}=\begin{pmatrix}9\\-10\\7\end{pmatrix}\)

    Check: its scalar product with each side should be 0.

  4. Magnitude

    \(\text{Area}=|\mathbf{a}\times\mathbf{b}|=\sqrt{230}=15.2\)

    For a parallelogram there is no half.

Video walkthrough coming
Height of a triangle from its area

The triangle ABC has \(A(0, 1, 2)\), \(B(3, 3, 0)\) and \(C(2, -1, 5)\). Find the shortest distance from C to the line AB.

hABC
Working
  1. The area

    \(\overrightarrow{AB}=\begin{pmatrix}3\\2\\-2\end{pmatrix}\), \(\overrightarrow{AC}=\begin{pmatrix}2\\-2\\3\end{pmatrix}\), \(\overrightarrow{AB}\times\overrightarrow{AC}=\begin{pmatrix}2\\-13\\-10\end{pmatrix}\)

    Find the area first, with the vector product.

  2. Area

    \(\text{Area}=\tfrac12\sqrt{273}=8.261\)

    Keep extra figures until the end.

  3. Base

    \(|\overrightarrow{AB}|=\sqrt{17}=4.123\)

    Take AB as the base. The distance from C to AB is the height.

  4. Height

    \(\tfrac12\times4.123\times h=8.261\), so \(h=4.01\)

    Area is half base times height. Make h the subject.

Video walkthrough coming
A triangular glass panel

A triangular glass panel has its corners at \(P(0, 0, 1)\), \(Q(4, 1, 3)\), \(R(1, 5, 2)\), in metres. Glass costs \$85 per square metre. Find the cost of the panel.

Working
  1. Two sides

    \(\overrightarrow{PQ}=\begin{pmatrix}4\\1\\2\end{pmatrix}\), \(\overrightarrow{PR}=\begin{pmatrix}1\\5\\1\end{pmatrix}\)

    Units are metres.

  2. Vector product

    \(\mathbf{a}\times\mathbf{b}=\begin{pmatrix}4\\1\\2\end{pmatrix}\times\begin{pmatrix}1\\5\\1\end{pmatrix}=\begin{pmatrix}1\times1-2\times5\\ 2\times1-4\times1\\ 4\times5-1\times1\end{pmatrix}=\begin{pmatrix}-9\\-2\\19\end{pmatrix}\)

  3. Area

    \(\tfrac12\sqrt{446}=10.56\ \text{m}^2\)

    Half of the magnitude.

  4. Cost

    \(\tfrac12\sqrt{446}\times85=\$897.55\)

    Use the unrounded area. Money to 2 decimal places.

Video walkthrough coming