IB Mathematics Applications & Interpretation AHL 3.10—vector Definitions Notes

Why Vectors Exist: Size And Direction In One Object

1. Some quantities are fully described by a single number, such as the mass of a parcel, the air temperature, or the time a delivery takes; these are scalars.
2. Other quantities make no sense until you also say which way, since a delivery drone that flies $5\,\mathrm{km}$ has gone nowhere useful until you know the direction.
3. Velocity, force, and displacement each carry a direction as well as a size.
4. A vector is a single object that stores a magnitude and a direction together, which is exactly what these quantities need.

Position And Displacement Vectors

1. A position vector pins a point to the origin, so the point $A$ has position vector $\vec{OA} = \mathbf{a}$, running from the origin $O$ out to $A$.
2. A displacement vector describes the trip from one point to another, so from $A$ to $B$ it is $\vec{AB}$, and the reverse trip is $\vec{BA} = -\vec{AB}$.
3. To find the displacement, follow the arrows $O \to A \to B$ tip-to-tail, which gives $\vec{OA} + \vec{AB} = \vec{OB}$.
   1. Rearranging isolates the vector between the two points: $$\vec{AB} = \vec{OB} - \vec{OA} = \mathbf{b} - \mathbf{a}$$
4. The distance between $A$ and $B$ is just the length of that displacement vector, $|\vec{AB}| = |\mathbf{b} - \mathbf{a}|$.
5. In the figure below, $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$ are the two position vectors, and the vector joining them is $\vec{AB} = \mathbf{b} - \mathbf{a}$.

Column Form, Component Form, And Base Vectors

1. A vector is written in column form by stacking its components, $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ in two dimensions and $\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$ in three.
2. The components $v_1, v_2, v_3$ tell you how far the vector reaches along the $x$-, $y$-, and $z$-axes.
3. There are three base vectors, each a unit step along one axis: $$\mathbf{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$
4. Any vector is a combination of these three, so the column form and the component form say the same thing: $$\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}$$

Magnitude And Distance Between Two Points

1. The magnitude $|\mathbf{v}|$ is the vector's length, and it comes straight from Pythagoras.
2. In two dimensions the components $v_1$ and $v_2$ are the legs of a right triangle, so $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}$.
3. In three dimensions apply Pythagoras twice, which extends the same idea to $$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$$

Distance Between Two Points

1. The distance between two points is the length of the vector joining them, $AB = |\mathbf{b} - \mathbf{a}|$.
2. So the method is always two steps: subtract the position vectors to get the displacement, then take its magnitude.

The Zero Vector, Negatives, And Scalar Multiplication

1. The zero vector $\mathbf{0}$ has every component zero and no direction, and it is the position vector of the origin.
2. The negative $-\mathbf{v}$ keeps the same magnitude but reverses the direction, flipping the sign of every component.
3. Multiplying by a scalar $k$ multiplies each component by $k$: $$k\mathbf{v} = \begin{pmatrix} kv_1 \\ kv_2 \\ kv_3 \end{pmatrix}$$
   1. If $k > 0$ the vector keeps its direction and is stretched, and if $k < 0$ it reverses, with the new length equal to $|k||\mathbf{v}|$.
4. This gives the test for parallel vectors, since two vectors point along the same line exactly when one is a scalar multiple of the other: $$\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = k\mathbf{b} \text{ for some scalar } k \neq 0$$

Adding And Subtracting Vectors

1. Add or subtract vectors component by component: $$\mathbf{a} \pm \mathbf{b} = \begin{pmatrix} a_1 \pm b_1 \\ a_2 \pm b_2 \\ a_3 \pm b_3 \end{pmatrix}$$
2. Geometrically, addition follows the triangle (tip-to-tail) rule: place the tail of $\mathbf{b}$ at the tip of $\mathbf{a}$, and the sum runs from the start of $\mathbf{a}$ to the end of $\mathbf{b}$.
3. The parallelogram rule is the same picture drawn from a shared start point, where the sum is the diagonal of the parallelogram.
4. Subtraction is just adding the negative, so $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$.

Unit Vectors And Normalizing

1. A unit vector has length $1$, so it carries a pure direction with no size attached.
2. To normalize any vector into a unit vector pointing the same way, divide it by its own magnitude: $$\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}$$

Putting It Together: A Modelling Question

1. A real AI question usually chains these steps together: find a displacement, measure a distance, then read off a direction.
2. Work in context and interpret each answer, since the numbers describe something physical.