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Basic operations

Addition and scalar multiplication of vectors

The scalar multiplication of a vector $\uvec$ by a number $\lambda$ is the vector $\lambda\uvec$ obtained by stretching the vector by a factor $\vert\lambda\vert$, and also flipping it if $\lambda\lt 0$.

More precisely, $\lambda \uvec$ is the vector of length $\vert \lambda\vert\, \vert \uvec\vert$. It has the same direction as $\uvec$ if $\lambda\ge0$ and the opposite direction if $\lambda\lt0$. The number $\lambda$ is often called a scalar in this context, to distinguish it from a vector.

The addition of two vectors $\uvec$ and $\vvec$ is the vector obtained by translating the start point of $\vvec$ to the endpoint of $\uvec$.

In a Cartesian coordinate system, scalar multiplication and addition are obtained by multiplication, resp. addition, of each of the coordinates \begin{gather*} \lambda(x,y,z) = (\lambda x, \lambda y, \lambda z),\\ (x_1,y_1,z_1) + (x_2,y_2,z_2) = (x_1+x_2,y_1+y_2,z_1+z_2). \end{gather*}

Addition and scalar multiplication in a coordinate system

Algebraic properties of addition and scalar multiplication of vectors

The usual algebraic properties valid for numbers are valid for vectors:

• Commutativity $\uvec+\vvec = \vvec+\uvec$
• Associativity $\uvec+(\vvec+\wvec) = (\uvec+\vvec)+\wvec$
• Distributivity $\lambda(\uvec +\vvec) = \lambda\uvec + \lambda\vvec$
Informally, commutativity says that vectors can be swapped; associativity says that order of operation doesn't matter; and distributivity says how brackets expand.

Collinearity

Two non-zero vectors are collinear or parallel if they have the same (unoriented) direction. In other words, they are proportional to each other by a scalar.

To verify that two vectors $\uvec$ and $\vvec\ne 0$ are collinear, it is necessary to find $\lambda\in\R$ such that $$\uvec = \lambda\vvec.$$

The vector $0$ is collinear to every vector.

Collinear vectors

Vector length and triangle inequality

The length, or magnitude of a vector is given by $$\vert \uvec\vert = \sqrt{x^2+y^2},\qquad \vert \vvec\vert = \sqrt{x^2+y^2+z^2}.$$

The triangle inequality states that the length of the sum of two vectors is smaller than the sum of the length of the vectors $$\vert \uvec+\vvec \vert \le \vert \uvec \vert + \vert \vvec\vert.$$

A unit vector is a vector of length one.

The triangle inequality for vectors

The ratio theorem

We say that $C$ divides $AB$ in a ratio $\mu:\lambda$ if $$\lambda\overrightarrow{CA} + \mu\overrightarrow{CB}=0.$$ When $\lambda$ and $\mu$ are positive, this is equivalent to saying that $C$ is in the segment $AB$ and that $\displaystyle \frac{d(C,A)}{d(C,B)}=\frac{\mu}{\lambda}$ (where $d(C,A)$ is the distance between $C$ and $A$).

The ratio theorem says that $C$ divides $AB$ in a ratio $\mu: \lambda$ if and only if $$\overrightarrow{OC} =\frac{\lambda\overrightarrow{OA} +\mu\overrightarrow{OB}}{\lambda +\mu}.$$

The midpoint divides $AB$ with ratio $1:1$. The midpoint $C$ is thus $$\overrightarrow{OC} =\frac{ \overrightarrow{OA} + \overrightarrow{OB}}{2}.$$

The ratio theorem