Vector algebra | Improve Society

The operations of addition, subtraction and multiplication familiar in the algebra of numbers are, with suitable definition, capable of extension to an algebra of vectors. The following definitions are fundamental.

Two vectors $\bold{A}$ and $\bold{B}$ are equal if they have the same magnitude and direction regardless of their initial points.

A vector having direction opposite to that of vector $\bold{A}$ but with the same magnitude is denoted by $-\bold{A}$ .

The sum or resultant of vectors $\bold{A}$ and $\bold{B}$ is a vector $\bold{C}$ formed by placing the initial point of $\bold{B}$ on the terminal point of $\bold{A}$ and joining the initial point of $\bold{A}$ to the terminal point of $\bold{B}$ . The sum $\bold{C}$ is written $\bold{C} = \bold{A} + \bold{B}$ . The definition here is equivalent to the parallelogram law for vector addition.

The difference of vectors $\bold{A}$ and $\bold{B}$ , represented by $\bold{A} - \bold{B}$ , is that vector $\bold{C}$ which added to $\bold{B}$ gives $\bold{A}$ . Equivalently, $\bold{A} - \bold{B}$ may be defined as $\bold{A} + (-\bold{B})$ . If $\bold{A} = \bold{B}$ , then $\bold{A} - \bold{B}$ is defined as the null or zero vector and is represented by the symbol $\bold{0}$ . This has a magnitude of zero but its direction is not defined.

Multiplication of vector $\bold{A}$ by a scalar m produces a vector $m\bold{A}$ with magnitude $|m|$ times the magnitude of $\bold{A}$ and direction the same as or opposite to that of $\bold{A}$ according as $m$ is positive or negative. If $m = 0$ , $m\bold{A} = \bold{0}$ , the null vector.