Component Vectors in 2D and 3D (SQA National 5 Maths): Revision Note

Vectors in 2 dimensions

What are vectors?

• A vector is a type of number that has both a size and a direction

• There are several different notations for vectors

  • Vectors may be indicated by bold lower-case letters

    • $\mathbf{a},\mathbf{b},\mathbf{r},\mathbf{s}$, etc.

    • In an exam you can underline the letter to show it is a vector

      • a, b, r, s, etc.

  • Vectors can be indicated by two upper-case letters with an arrow on top

    • $\overrightarrow{\mathrm{AB}}$ is the vector that starts at point $\mathrm{A}$ and ends at point $\mathrm{B}$

    • $\overrightarrow{\mathrm{BA}}$ is the vector that starts at point $\mathrm{B}$ and ends at point $\mathrm{A}$

    • $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{BA}}$ are not equal

      • They have the same size

      • But different directions (one points from $\mathrm{A}$ to $\mathrm{B}$, and the other points from $\mathrm{B}$ to $\mathrm{A}$

• Vectors can also be written in component form

  • This indicates precise distances and directions in two dimensions or three dimensions

How do I write a two-dimensional vector in component form?

• A vector in component form can be used to describe how to get from one point to another point

  • The two-dimensional vector $\left(\begin{array}{c}6\\ 3\end{array}\right)$ means

    • 6 units to the right (i.e. in the positive x direction)

    • and 3 units up (i.e. in the positive y direction)

How do I find the component form of a vector between two points?

• To find a vector between two points in component form:

  • E.g. vector $\overrightarrow{\mathrm{AB}}$, for points $\mathrm{A}\left(2,-3\right)$ and $\mathrm{B}\left(7,4\right)$

  • Subtract the coordinates for the first point ($\mathrm{A}$) from the coordinates for the second point ($\mathrm{B}$)

    • $\overrightarrow{\mathrm{AB}}=\left(\begin{array}{c}7-2\\ 4-\left(-3\right)\end{array}\right)=\left(\begin{array}{c}5\\ 7\end{array}\right)$

  • You can check that this makes sense

    • Going 5 to the right and 7 up from $\left(2,-3\right)$ does indeed get you to $\left(7,4\right)$

• If you reverse the order of the points, the components of the vector change sign

  • E.g. vector $\overrightarrow{\mathrm{BA}}$, for points $\mathrm{A}\left(2,-3\right)$ and $\mathrm{B}\left(7,4\right)$

    • $\overrightarrow{\mathrm{BA}}=\left(\begin{array}{c}2-7\\ -3-4\end{array}\right)=\left(\begin{array}{c}-5\\ -7\end{array}\right)$

    • $\overrightarrow{\mathrm{AB}}\ne \overrightarrow{\mathrm{BA}}$

Vectors and coordinates in 3 dimensions

What are 3D coordinates?

• Vectors in component form can also be used in three dimensions

• Normally the three dimensions are labelled x, y, and z

  • The z-axis is perpendicular to the x- and y-axes

  • If you draw standard two-dimensional x- and y-axes on a sheet of paper

    • then the z-axis will go through the origin, perpendicular to the sheet of paper

    • The positive z direction will be up out of the sheet of paper

• The coordinate point (x, y, z) indicates

  • the distance from the origin along the x-axis

  • the distance from the origin along the y-axis

  • and the distance from the origin along the z-axis

How do I write a three-dimensional vector in component form?

• In three dimensions, a vector in component form has a third number to indicate distances in the z direction

  • The three-dimensional vector $\left(\begin{array}{c}2\\ -3\\ 4\end{array}\right)$ means

    • 2 units in the positive x direction

    • 3 units in the negative y direction

    • and 4 units in the positive z direction

• Finding the component form of a vector between two points works exactly the same as in two dimensions

  • E.g. vector $\overrightarrow{\mathrm{PQ}}$, for points $\mathrm{P}\left(3,4,-2\right)$ and $\mathrm{Q}\left(-1,6,-1\right)$

  • Subtract the coordinates for the first point ($\mathrm{P}$) from the coordinates for the second point ($\mathrm{Q}$)

    • $\overrightarrow{\mathrm{PQ}}=\left(\begin{array}{c}-1-3\\ 6-4\\ -1-\left(-2\right)\end{array}\right)=\left(\begin{array}{c}-4\\ 2\\ 1\end{array}\right)$