# 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

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

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

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

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

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

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

A Vector sum of 2 vectors

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

Example

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

## Substraction of 2 vector

A Vector difference of 2 vectors

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

Example

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

## Multiplication by a scalar

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

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

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