AHL 3.12—Vector definitions Notes

Vector Concepts and Representations

Vectors are fundamental mathematical objects that possess both magnitude and direction, describing a translation. Hence they showcase both quantity and orientation in some space.

They can be written as a bold letter or with a vector sign on top, for example

$$\vec{v} \text{ OR } \mathbf{v}$$

Position and Displacement Vectors

Position vectors represent the location of a point in space relative to a fixed origin. For instance, if we consider a point P in three-dimensional space, its position vector $\vec{OP}$ would extend from the origin O to the point P.

Displacement vectors, on the other hand, describe the change in position from one point to another. If we have two points A and B, the displacement vector $\vec{AB}$ represents the directed path from A to B. Or from B to A would be $\vec{BA}$

Let the right red position vector be $\vec{OA}=\mathbf{a}$ and the left red position vector be $\vec{OB} = \mathbf{b}$. Then the blue vector goes from $\vec{OA}$ to $\vec{OB}$ which is $\vec{AB}$ and can be calculated as $\mathbf{b}- \mathbf{a}$

Representation Using Directed Line Segments

Vectors are often visually represented as arrows, where the length of the arrow corresponds to the vector's magnitude, and the direction of the arrow indicates the vector's direction.

Base Vectors and Components

Base Vectors i, j, k

In three-dimensional space, we define three standard unit vectors:

• $i = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$
• $j = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$
• $k = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$

These vectors form a basis for 3D space, meaning any vector can be expressed as a linear combination of these base vectors.

Components of a Vector

A vector v can be expressed in terms of its components:

$\mathbf{v} = (v_1, v_2, v_3) = v_1\times i + v_2\times j + v_3\times k$

Where $v_1$, $v_2$, and $v_3$ are scalar values representing the vector's components along the x, y, and z axes respectively.

Magnitude and Unit

Since a vector goes in multiple directions, the actual length of the vector, or in other words the magnitude, can be calculated by taking the sum of the squares of each component. This is a simple property of Pythagoras, and for 2-dimensional or 3-dimensional, can be easily shown.

The magnitude is written as $||\mathbf{v}||$ and for a vector $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$ can be calculated as

$$||\mathbf{v}|| = \sqrt{v_1^2 + v_2^2 + v_3^2}$$

A vector that has a length of $1$ is called a unit vector. In fact a vector going one direction can be scaled into a unit vector in the same direction by

$$\mathbf{u} = \frac{\mathbf{v}}{||\mathbf{v}||}$$