We define a vector as a entity involving both magnitude and direction. A vector can be represented by an arrow. In this section the geometry of vectors is discussed, and it is derived from physical experiments on how two vector quantities interact.
Physical experiments show that if two like quantities act together, their effect is predictable. In this case, the vectors used to represent these quantities can be combined to form a resultant vector that represents the combined effects of the original quantities, which is called the sum. The rule for their combination is called the parallelogram law.
Parallelogram Law for Vector Addition. The sum of two vectors $x$ and $y$ that act at the same point $P$ is the vector beginning at $P$ that is represented by the diagonal of parallelogram having $x$ and $y$ as adjacent sides.
Vector addition can be described algebraically with the use of analytic geometry. We introduce a plane with the Cartesian coordinates system. Let $(a_1,a_2)$ denote the endpoint of $x$ and $(b_1,b_2)$ denote the endpoint of $y$. Then the end point of $x+y$ is $(a_1+b_1,a_2+b_2)$.
There is another natural operation that can be performed on vectors—scalar multiplication. If a vector $x$ is represented by an arrow, then for any nonzero real number $t$, the vector $tx$ is represented by an arrow in the same direction if $t>0$ and in the opposite direction if $t<0$. And, the length of $tx$ is $\vert t \vert$ times the length of $x$. Two nonzero vectors $x$ and $y$ are called parallel if $y=tx$ for some nonzero real number $t$.
The algebraic descriptions of vector addition and scalar multiplication yield the following properties:
- For all vectors $x$ and $y$, $x+y=y+x$.
- For all vectors $x,y,$ and $z, (x+y)+z=x+(y+z)$.
- There exists a vector denoted $0$ such that $x+0=x$ for each vector $x$.
- For each vector $x$, there exists a vector $y$ such that $x+y=0$.
- For each vector $x$, $1x=x$.
- For each pair of real numbers $a$ and $b$ and each vector $x$, $(ab)x=a(bx)$.
- For each pair of real numbers $a$ and $b$ and each vector $x$, $(a+b)x=ax+bx$.
- For each real number $a$ and each pair of vectors $x$ and $y$, $a(x+y)=ax+ay$.