Menu
Sign up Sign in

AI HL · AHL 4

Vectors

0 of 160 skills done

0
AHL 4.2

Adding and subtracting vectors

AHL 4.2

Adding and subtracting vectors

Target: I can add and subtract column vectors, in 2D and 3D, component by component. (+4 more)

By the end of this section

Key points

Adding vectors

  • To add on a diagram, join the tail of the second vector to the head of the first. The sum goes from the first tail to the second head. It is called the resultant.
  • Add the components in a column: \(\begin{pmatrix}3\\1\end{pmatrix}+\begin{pmatrix}2\\4\end{pmatrix}=\begin{pmatrix}5\\5\end{pmatrix}\).
  • The sum is also the diagonal of the parallelogram built on the two vectors.
  • Equal vectors have the same size and direction, wherever they start.

Subtracting vectors

  • The negative \(-\mathbf{a}\) has the same length as \(\mathbf{a}\) and the opposite direction. Every component changes sign.
  • Subtracting is adding the negative: \(\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})\).
  • The order matters. \(\mathbf{b}-\mathbf{a}=-(\mathbf{a}-\mathbf{b})\).
  • \(\mathbf{a}-\mathbf{a}=\mathbf{0}\), the zero vector.

Vectors between points

  • \(\overrightarrow{AB}\) is the vector from A to B, and its length is the length AB.
  • \(\overrightarrow{BA}=-\overrightarrow{AB}\): swapping the end points reverses the vector.
  • To go from one point to another, follow any route along the given vectors: \(\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}\).
  • Travelling with an arrow means add it. Travelling against an arrow means subtract it.

Explore: adding vectors head to tail

Drag the tip of a (black) and the tip of b (teal). The orange arrow is the sum. Predict it before you read the working.

The points snap to whole numbers. Select one (Tab) and use the arrow keys to move it.

  1. Drag the tip of b so that a + b is the zero vector. What is b in terms of a?
  2. Does b + a give the same arrow as a + b? Swap the two vectors in your head and check the column.
  3. Show a − b. Now change b and watch both ends of the arrow. Why does the column for a − b not match the one for b − a?
  4. The arrow for a + b is a diagonal of a parallelogram. Which two sides make the other diagonal?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Adding column vectors

Given \(\mathbf{u}=\begin{pmatrix}2\\-5\\3\end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix}4\\1\\-6\end{pmatrix}\), find the resultant \(\mathbf{u}+\mathbf{v}\).

Working
  1. Add component by component

    \(\mathbf{u}+\mathbf{v}=\begin{pmatrix}2+4\\-5+1\\3+(-6)\end{pmatrix}\)

    Top with top, middle with middle, bottom with bottom.

  2. The resultant

    \(\mathbf{u}+\mathbf{v}=\begin{pmatrix}6\\-4\\-3\end{pmatrix}\)

    The sum of vectors is called the resultant. On a diagram it joins the tail of u to the head of v when v starts where u ends.

Video walkthrough coming
Subtracting and the order

Given \(\mathbf{a}=\begin{pmatrix}7\\2\\-4\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}3\\5\\-1\end{pmatrix}\), find \(\mathbf{a}-\mathbf{b}\) and \(\mathbf{b}-\mathbf{a}\).

Working
  1. Add the negative

    \(\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})=\begin{pmatrix}7\\2\\-4\end{pmatrix}+\begin{pmatrix}-3\\-5\\1\end{pmatrix}\)

    Subtracting b is the same as adding −b, which has every component reversed in sign.

  2. Add components

    \(\mathbf{a}-\mathbf{b}=\begin{pmatrix}4\\-3\\-3\end{pmatrix}\)

    Do each row on its own.

  3. Swap the order

    \(\mathbf{b}-\mathbf{a}=\begin{pmatrix}-4\\3\\3\end{pmatrix}\)

    b − a is −(a − b): every sign changes. The order matters in a subtraction.

Video walkthrough coming
Along a path

In the diagram, \(\overrightarrow{PQ}=\mathbf{u}\), \(\overrightarrow{QR}=\mathbf{v}\) and \(\overrightarrow{RS}=\mathbf{w}\). Write \(\overrightarrow{PS}\) in terms of \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\).

uvwPQRS
Working
  1. Choose a route

    \(\overrightarrow{PS}=\overrightarrow{PQ}+\overrightarrow{QR}+\overrightarrow{RS}\)

    Go from P to S along the arrows. Each arrow starts where the last one ends.

  2. Replace each arrow

    \(\overrightarrow{PS}=\mathbf{u}+\mathbf{v}+\mathbf{w}\)

    When you travel with an arrow, add it.

Video walkthrough coming
Against the arrows

Use the same diagram. Write \(\overrightarrow{SQ}\) in terms of \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\).

uvwPQRS
Working
  1. The forward path

    \(\overrightarrow{QS}=\overrightarrow{QR}+\overrightarrow{RS}=\mathbf{v}+\mathbf{w}\)

    Q to S follows the arrows, so write that first.

  2. Reverse it

    \(\overrightarrow{SQ}=-\overrightarrow{QS}=-(\mathbf{v}+\mathbf{w})=-\mathbf{v}-\mathbf{w}\)

    A reversed vector is the negative: BA = −AB. Every sign changes.

Video walkthrough coming