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Basic operations

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
Addition and scalar multiplication in a coordinate system

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.

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
Collinear vectors

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 triangle inequality for vectors

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
The ratio theorem