IB Maths AI 3.9 Notes

Scalars & Vectors

A scalar only tells us size. A vector tells us size and direction.

For example, speed might be $10$ m s$^{-1}$, but velocity must also include direction. If an object moves north at $10$ m s$^{-1}$ and then south at $10$ m s$^{-1}$, the speed is the same in both cases, but the velocities are different.

Vectors can be shown as arrows. The arrow points in the direction of the vector, and its length represents the magnitude. They can also be written in component form.

• In two dimensions, $\mathbf{v}=x\mathbf{i}+y\mathbf{j}=(x,y) = \begin{pmatrix}x\\y\end{pmatrix}$
• In three dimensions, $\mathbf{v}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}=(x,y,z)=\begin{pmatrix}x\\y\\z\end{pmatrix}$

The base vectors are:

• $\mathbf{i}$ for one unit in the $x$-direction
• $\mathbf{j}$ for one unit in the $y$-direction
• $\mathbf{k}$ for one unit in the $z$-direction