Geometric Vectors and Vector Addition

Definition The directed line segment from a point $P$ to a point $Q$ is the line segment between them together with a direction, from its initial point $P$ to its terminal point $Q$. It is written $\overrightarrow{PQ}$ and drawn as an arrow with its head at $Q$. Its length is the distance between $P$ and $Q$. The segments $\overrightarrow{PQ}$ and $\overrightarrow{QP}$ use the same two points but are different, because they point opposite ways.

Definition Two directed line segments are equivalent if they have the same length and point in the same direction, and all segments of length zero are equivalent to each other. A vector is a quantity that has a magnitude and a direction, with one exception, the zero vector $\mathbf{0}$, which has magnitude $0$ and no direction. Geometrically a vector is represented by a directed line segment, and equivalent segments represent the same vector, so a vector may be drawn starting at any point. The zero vector is represented by a segment from a point to itself. Vectors are written in bold, such as $\mathbf{v}$, or as $\overrightarrow{PQ}$ for the vector that the segment from $P$ to $Q$ represents. A quantity with a size but no direction, such as a mass or a temperature, is called a scalar.

Definition The magnitude of a vector $\mathbf{v}$, written $\lVert \mathbf{v} \rVert$ (some books write $\lvert \mathbf{v} \rvert$), is the length of any directed line segment that represents it. Equivalent segments have equal lengths, so the choice of segment does not matter. The magnitude is never negative, and it is $0$ only for the zero vector. If $\mathbf{v} = \overrightarrow{PQ}$ with $P = (x_1, y_1)$ and $Q = (x_2, y_2)$, the distance formula gives

Definition Two nonzero vectors have the same direction when their segments are parallel, or lie on one line, and point the same way. They have opposite directions when their segments are parallel but point opposite ways. In the coordinate plane the direction of a nonzero vector $\mathbf{v}$ is usually given by its direction angle $\theta$. When $\mathbf{v}$ is drawn with its initial point at the origin, $\theta$ is the angle from the positive $x$-axis to $\mathbf{v}$, measured counterclockwise, with $0^\circ \le \theta < 360^\circ$. The zero vector has no direction.

Definition To add vectors $\mathbf{u}$ and $\mathbf{v}$ head to tail, also called tip to tail, draw $\mathbf{u}$, then draw $\mathbf{v}$ with its initial point at the terminal point of $\mathbf{u}$. The sum $\mathbf{u} + \mathbf{v}$, also called the resultant, is the vector from the initial point of $\mathbf{u}$ to the terminal point of $\mathbf{v}$. In terms of points,

Vector addition is associative, so a longer sum is drawn by chaining all the arrows head to tail. Magnitudes satisfy the triangle inequality $\lVert \mathbf{u} + \mathbf{v} \rVert \le \lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$, with equality exactly when one of the vectors is zero or both have the same direction.

Definition Let $\mathbf{u} = \overrightarrow{PQ}$ and $\mathbf{v} = \overrightarrow{PS}$ be drawn from the same initial point $P$, not along one line, and complete the parallelogram $PQRS$. Then the diagonal from $P$ is the sum,

The opposite sides $\overrightarrow{QR}$ and $\overrightarrow{SR}$ represent $\mathbf{v}$ and $\mathbf{u}$, so going around either side gives the same diagonal, and $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$. If $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$ at $P$, with $0^\circ < \theta < 180^\circ$, then the law of cosines in triangle $PQR$, whose angle at $Q$ is $180^\circ - \theta$, gives

The same formula holds when the vectors lie along one line, with $\theta = 0^\circ$ or $\theta = 180^\circ$.

Definition The opposite of a vector $\mathbf{v}$, written $-\mathbf{v}$, has the same magnitude as $\mathbf{v}$ and the opposite direction, and $-\mathbf{0} = \mathbf{0}$. If $\mathbf{v} = \overrightarrow{PQ}$ then $-\mathbf{v} = \overrightarrow{QP}$, and $\mathbf{v} + (-\mathbf{v}) = \overrightarrow{PP} = \mathbf{0}$. Subtraction is defined by

When $\mathbf{u}$ and $\mathbf{v}$ are drawn from the same initial point, $\mathbf{u} - \mathbf{v}$ is the vector from the terminal point of $\mathbf{v}$ to the terminal point of $\mathbf{u}$, the other diagonal of the parallelogram.