MATH 20C Lecture 1

Vectors as arrows

Vectors can be represented as arrows from P to Q: or
In this example, P is the tail and Q is the head of the arrow.
The arrow has length and direction.
The length of a vector is denoted .

Relations between vectors

Parallel vectors

vectors with the same angle
can have opposite directions
assume that they are not the zero vector

Translations

ame direction and same length
” is a translation of “
In this class,MATH_20C , we consider and to be the same.

Vector operations

Addition

is defined as the diagonal of the parallelogram composed of , , and the parallels of the two vectors
can also be defined as the vector formed from the long leg of the triangle with and , where the head of is attached to the tail of .

Scalar multiplication

real number/scalar
vector
scalar multiplication is defined by stretching the length of the vector by .

, which is the unique vector with the same length with opposite direction.

In general, and are parallel iff

Properties of scalar multiplication

$||\lambda \vec{v}||=|\lambda ||\vec{v}||$

$\lambda \vec{v}$ points in the direction as $\vec{v}$ if $\lambda >0$, the oposite direction to $\vec{v}$ if $\lambda <0$

$0\vec{v}=\vec{0}$

$(-1)\vec{v}=-\vec{v}$

Vector subtraction

Algebraic Description