Vectors Simply Explained

Vector \( \class{red}{\boldsymbol{a}} \) represents a quantity described by a magnitude (length) \( \class{red}{a} \) and a direction. When written on paper, the vector is depicted with an arrow above the vector symbol \( \class{red}{\vec{a}} \). In printed form, this notation is unnecessary, and the vector is represented by a bold letter, as here on this page. The magnitude of the vector is sometimes represented by two vertical bars \( |\class{red}{\boldsymbol{a}}| \). Here, it is simply shown as non-bold: \( \class{red}{a} \).

Let us consider a three-dimensional coordinate system \((x,~y,~z)\). We can draw a three-dimensional vector in this coordinate system: \( \class{red}{\boldsymbol{a}} = (a_1,~a_2,~a_3) \). Three-dimensional means that the vector has three components \( a_1\), \(a_2\) and \(a_3\).

Here \(a_1\) is the length of the vector \( \class{red}{\boldsymbol{a}} \) in the \(x\)-direction, \(a_2\) is the length of the vector \( \class{red}{\boldsymbol{a}} \) in the \(y\)-direction and \(a_3\) is the length of the vector \( \class{red}{\boldsymbol{a}} \) in the \(z\)-direction.

We can do math with vectors. We can multiply a vector \( \class{red}{\boldsymbol{a}} \) by a number: \( 5\class{red}{\boldsymbol{a}} \). We can add two vectors \( \class{red}{\boldsymbol{a}} \) and \( \class{blue}{\boldsymbol{b}} \) \( \class{red}{\boldsymbol{a}} + \class{blue}{\boldsymbol{b}} \) and subtract \( \class{red}{\boldsymbol{a}} - \class{blue}{\boldsymbol{b}} \). We can even form a scalar product \( \class{red}{\boldsymbol{a}} \cdot \class{blue}{\boldsymbol{b}} \) and a cross product \( \class{red}{\boldsymbol{a}} \times \class{blue}{\boldsymbol{b}} \).