# Calculus II/Vector Algebra Operation

## Definitions of a vector

A vector is a mathematical concept that has both magnitude and direction. A vector is an object that has certain properties:

• Magnitude (or length) denoted as A
• Direction . denoted as a

A vector ${\vec {X}}$  of length X pointing from left to right horizontally

${\vec {X}}=X{\vec {x}}$

A vector ${\vec {Y}}$  of length X pointing from bottom to top vertically

${\vec {Y}}=Y{\vec {y}}$

From above, A vector ${\vec {X}}$  of length X pointing from left to right horizontally

${\vec {A}}=A{\vec {a}}$
$A={\frac {\vec {A}}{\vec {a}}}$
${\vec {a}}={\frac {\vec {A}}{A}}$

A Vector sum of 2 vectors

${\vec {Z}}={\vec {X}}+{\vec {Y}}=X{\vec {x}}+Y{\vec {y}}$

Example

${\vec {Z}}={\vec {X}}+{\vec {Y}}=3{\vec {x}}+2{\vec {y}}$

## Substraction of 2 vector

A Vector difference of 2 vectors

${\vec {Z}}={\vec {X}}-{\vec {Y}}=X{\vec {x}}-Y{\vec {y}}=X{\vec {x}}+(-Y{\vec {y}})$

Example

${\vec {Z}}={\vec {X}}-{\vec {Y}}=3{\vec {x}}-2{\vec {y}}=3{\vec {x}}+(-2{\vec {y}})$

## Multiplication by a scalar

Multiplication of a vector $\mathbf {b} \,$  by a scalar $\lambda \,$  has the effect of stretching or shrinking the vector (see Figure 2(b)).

You can form a unit vector ${\mathbf {\hat {b}} }\,$  that is parallel to $\mathbf {b} \,$  by dividing by the length of the vector $|\mathbf {b} |\,$ . Thus,

${\mathbf {\hat {b}} }={\cfrac {\mathbf {b} }{|\mathbf {b} |}}~.$