A vector has magnitude and direction. A column records horizontal then vertical displacement; i is one unit right and j is one unit up. OA= 4 3 =4 i+3 j. Translating the arrow without turning or stretching it gives the same vector. AO=- OA= -4 -3. Reversing an arrow changes the vector, but not its length. Add vectors head-to-tail or component by component: 4 3 + -1 2 = 3 5. Subtraction adds the opposite vector. Multiplication by a positive scalar stretches an arrow; a negative scalar also reverses its direction. -2 4 3 = -8 -6.

A vector has magnitude and direction. A column records horizontal then vertical displacement; \(\mathbf i\) is one unit right and \(\mathbf j\) is one unit up.

\(\overrightarrow{OA}=\begin{pmatrix}4\\3\end{pmatrix}=4\mathbf i+3\mathbf j\). Translating the arrow without turning or stretching it gives the same vector.

\(\overrightarrow{AO}=-\overrightarrow{OA}=\begin{pmatrix}-4\\-3\end{pmatrix}\). Reversing an arrow changes the vector, but not its length.

Add vectors head-to-tail or component by component: \(\begin{pmatrix}4\\3\end{pmatrix}+\begin{pmatrix}-1\\2\end{pmatrix}=\begin{pmatrix}3\\5\end{pmatrix}\). Subtraction adds the opposite vector.

Multiplication by a positive scalar stretches an arrow; a negative scalar also reverses its direction. \(-2\begin{pmatrix}4\\3\end{pmatrix}=\begin{pmatrix}-8\\-6\end{pmatrix}\).

Vectors and their components

Vectors and their components