Vectors Fundamentals

Vectors are usually written as a letter with a little arrow over it. The individual components of the vector are usually the same letter but without an arrow and a subscript numbering the components. Ways to write out the individual components are between angle brackets, as a column vector, and not as commonly as with the $\vec{i}$, $\vec{j}$, and $\vec{k}$ vectors.

In two dimensional space,

\begin{equation} \vec{a} = \langle a_{1},a_{2} \rangle = \begin{bmatrix} a_{1} \\ a_{2} \end{bmatrix} = a_{1}\vec{i} + a_{2}\vec{j} \end{equation} \begin{equation} \vec{i} = \langle 1,0 \rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \qquad \vec{j} = \langle 0,1 \rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{equation}

In three dimensional space,

\begin{equation} \vec{a} = \langle a_{1},a_{2},a_{3} \rangle = \begin{bmatrix} a_{1} \\ a_{2} \\ a_{3} \end{bmatrix} = a_{1}\vec{i} + a_{2}\vec{j} + a_{3}\vec{k} \end{equation} \begin{equation} \vec{i} = \langle 1,0,0 \rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \qquad \vec{j} = \langle 0,1,0 \rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \qquad \vec{k} = \langle 0,0,1 \rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \end{equation}

Vectors can be multiplied by scalars, which are just regular number. A scalar $c$ multiplying a vector $\vec{a}$ multiplies each of the components of $\vec{a}$.

\begin{equation} c \vec{a} = \langle ca_{1},ca_{2} \rangle = \begin{bmatrix} ca_{1} \\ ca_{2} \end{bmatrix} \qquad c \vec{a} = \langle ca_{1},ca_{2},ca_{3} \rangle = \begin{bmatrix} ca_{1} \\ ca_{2} \\ ca_{3} \end{bmatrix} \end{equation}

Vectors can be added each other to create another vector.

\begin{equation} \vec{a} + \vec{b} = \langle a_{1}+b_{1},a_{2}+b_{2} \rangle = \begin{bmatrix} a_{1}+b_{1} \\ a_{2}+b_{2} \end{bmatrix} \end{equation} \begin{equation} \vec{a} + \vec{b} = \langle a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3} \rangle = \begin{bmatrix} a_{1}+b_{1} \\ a_{2}+b_{2} \\ a_{3}+b_{3} \end{bmatrix} \end{equation}

The magnitude of a vector is just the length of the vector using the Pythagorean theorem in two dimensional space. The magnitude is commonly denoted by vertical bars, but there are many notations.

In three dimensional space, the magnitude is still determined by the Pythagorean theorem but requires two steps. In the following diagram, the length of the components in the $xy$-plane is computer first, which is used as the length of the base of a right triangle with height $a_{3}$.

\begin{equation} |\vec{a}| = \sqrt{\left(\sqrt{a_{1}^{2}+a_{2}^{2}}\right)^{2} + a_{3}^{2}} = \sqrt{a_{1}^{2}+a_{2}^{2} + a_{3}^{2}} \end{equation}