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AI HL · AHL 4

Vectors

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AHL 4.1

Column vectors and base vectors i, j, k

AHL 4.1

Column vectors and base vectors i, j, k

Target: I can say whether a quantity is a vector (size and direction) or a scalar (size only). (+4 more)

By the end of this section

Key points

Vectors and scalars

  • A vector has size (magnitude) and direction. Displacement, velocity and force are vectors.
  • A scalar has size only. Distance, speed, time, mass and energy are scalars.
  • A vector is written in bold, \(\mathbf{a}\), or underlined by hand. It is drawn as an arrow: the length is the size and the arrow shows the direction.
  • The start of the arrow is the tail and the end is the head.

Column vectors

  • \(\begin{pmatrix}3\\2\end{pmatrix}\) means 3 units to the right and 2 units up. The top number is across, the next is up or down.
  • Left and down are negative. \(\begin{pmatrix}-3\\-1\end{pmatrix}\) is 3 left and 1 down.
  • A 3D vector has three numbers in the column.
  • Equal vectors have equal components. Where they start does not matter.

Base vectors i, j and k

  • \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) each have length 1 and are perpendicular. \(\mathbf{i}\) points along the x-axis, \(\mathbf{j}\) along the y-axis and \(\mathbf{k}\) along the z-axis.
  • \(\begin{pmatrix}3\\2\\4\end{pmatrix}=3\mathbf{i}+2\mathbf{j}+4\mathbf{k}\). The numbers in the column are the coefficients of i, j and k.
  • A term that is missing means a component of 0: \(4\mathbf{i}-2\mathbf{k}=\begin{pmatrix}4\\0\\-2\end{pmatrix}\).
  • A column always goes in the order i, j, k, even if the terms are written in another order.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Reading vectors from a grid

Each grid square is one unit. Write the vectors \(\mathbf{m}\), \(\mathbf{n}\) and \(\mathbf{r}\) as column vectors. The black dot is the tail and the orange dot is the head.

mnr
Working
  1. Vector m

    \(\mathbf{m}=\begin{pmatrix}5\\2\end{pmatrix}\)

    Start at the tail. Count 5 units right, then 2 units up. Right and up are positive.

  2. Vector n

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

    3 units left is negative: −3. Then 4 units up: +4.

  3. Vector r

    \(\mathbf{r}=\begin{pmatrix}-2\\-3\end{pmatrix}\)

    2 units left and 3 units down: both components are negative.

  4. Using i and j

    \(\mathbf{m}=5\mathbf{i}+2\mathbf{j},\quad \mathbf{n}=-3\mathbf{i}+4\mathbf{j},\quad \mathbf{r}=-2\mathbf{i}-3\mathbf{j}\)

    The same vectors written with base vectors: the numbers in the column are the coefficients of i and j.

Video walkthrough coming
Column vectors to i, j, k

Write each vector using the base vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\): \(\mathbf{a}=\begin{pmatrix}6\\-4\end{pmatrix}\), \(\mathbf{b}=\begin{pmatrix}-2\\3\\7\end{pmatrix}\), \(\mathbf{c}=\begin{pmatrix}5\\0\\-9\end{pmatrix}\).

Working
  1. A 2D vector

    \(\mathbf{a}=6\mathbf{i}-4\mathbf{j}\)

    Top number goes with i, second with j. A 2D vector has no k.

  2. A 3D vector

    \(\mathbf{b}=-2\mathbf{i}+3\mathbf{j}+7\mathbf{k}\)

    Three numbers: coefficients of i, j and k in that order. A coefficient of −1 is written as just −i.

  3. A zero component

    \(\mathbf{c}=5\mathbf{i}-9\mathbf{k}\)

    The middle number is 0, so there is no j term. Leave it out.

Video walkthrough coming
i, j, k to column vectors

Write each vector as a column vector with three components: \(\mathbf{d}=3\mathbf{i}-5\mathbf{j}+2\mathbf{k}\), \(\mathbf{e}=4\mathbf{j}-7\mathbf{k}\), \(\mathbf{f}=8\mathbf{k}-2\mathbf{i}\).

Working
  1. All three present

    \(\mathbf{d}=\begin{pmatrix}3\\-5\\2\end{pmatrix}\)

    Read off the coefficients of i, j and k.

  2. A missing i

    \(\mathbf{e}=\begin{pmatrix}0\\4\\-7\end{pmatrix}\)

    e has no i term, so its i-component is 0.

  3. Terms in a different order

    \(\mathbf{f}=\begin{pmatrix}-2\\0\\8\end{pmatrix}\)

    f is written k first, but a column always goes i, j, k. There is no j term, so the middle number is 0.

Video walkthrough coming