Home page for accesible maths 1 Vectors 1.2 Scalar multiplication and unit vectors 1.4 The scalar product

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1.3 Addition of vectors

If a body moves from $A$ to $B$ , then from $B$ to $C$ , it has moved from $A$ to $C$ ! We define addition of vectors (represented as line segments) in this way: $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$ .

Now let $A, B$ be the points with position vectors a, b. The point with position vector ${\bf a}+{\bf b}$ is found as follows. Let $A C$ be parallel to $O B$ , with the same length, so that $O A C B$ is a parallelogram. Then $C$ is the required point, since

{\bf a}+{\bf b}=\overrightarrow{OA}+\overrightarrow{OB}=\overrightarrow{OA}+% \overrightarrow{AC}=\overrightarrow{OC}.

Let ${\bf a}=(x_{1}\;\;\ldots\;\;x_{n})$ and ${\bf b}=(y_{1}\;\;\ldots\;\;y_{n})$ . When we move along line segments corresponding to ${\bf a}$ and ${\bf b}$ , we add $x_{1}$ , then $y_{1}$ to the first coordinate. By doing both, we add $x_{1}+y_{1}$ . So in component form, the obvious formula applies:

(x_{1}\;\;\ldots\;\;x_{n})+(y_{1}\;\;\ldots\;\;y_{n})=(x_{1}+y_{1}\;\;\ldots\;% \;x_{n}+y_{n}).

Clearly, $\lambda({\bf a}+{\bf b})=\lambda{\bf a}+\lambda{\bf b}$ .

We write (of course) ${\bf a}-{\bf b}$ for ${\bf a}+(-{\bf b})$ . If $A,\,B$ have position vectors ${\bf a},\,{\bf b}$ , then $\overrightarrow{AB}={\bf b}-{\bf a}$ , since $\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}$ , in other words ${\bf a}+\overrightarrow{AB}={\bf b}$ .