PX153 - A2 - cartesian coordinates and vector components

basis vectors

3 vectors that are linearly independent (don't lie in the same plane $α \underset{―}{e_{1}} + β \underset{―}{e_{2}} + γ \underset{―}{e_{3}} \neq 0$ for non-zero $α, β, γ$ )
- then $\underset{―}{a} = a_{1} \underset{―}{e_{1}} + a_{2} \underset{―}{e_{2}} + a_{3} \underset{―}{e_{3}}$
- $\underset{―}{e_{1}}, \underset{―}{e_{2}}, \underset{―}{e_{3}}$ form a basis
- $a_{1}, a_{2}, a_{3}$ are the components of $\underset{―}{a}$ in that basis
normally orthonormal basis vectors are chosen
- orthonormal = perpendicular vectors with length one
- eg: cartesian coordiantes: $\underset{―}{{\hat{e}}_{x}}, \underset{―}{{\hat{e}}_{y}}, \underset{―}{{\hat{e}}_{z}}$

\vec{v} = (\underset{―}{{\hat{v}}_{x}}, \underset{―}{{\hat{v}}_{y}}, \underset{―}{{\hat{v}}_{z})} = v_{x} \underset{―}{{\hat{e}}_{x}} + v_{2} \underset{―}{{\hat{e}}_{y}} + v_{3} \underset{―}{{\hat{e}}_{z}} = \sum_{i = x, y, z} v_{i} \underset{―}{{\hat{e}}_{i}}

| v | = \sqrt{{v_{x}}^{2} + {v_{y}}^{2} + {v_{z}}^{2}}

the direction of $\vec{v}$

\underset{―}{\hat{v}} = \frac{v_{x}}{| \vec{v} |} \underset{―}{{\hat{e}}_{x}} + \frac{v_{y}}{| \vec{v} |} \underset{―}{{\hat{e}}_{y}} + \frac{v_{z}}{| \vec{v} |} \underset{―}{{\hat{e}}_{z}}

the value of each component of $\underset{―}{v}$ , is the projection of $\underset{―}{v}$ along the appropriate basis vector

we can define the direction cosines of $\underset{―}{v}$ which relate to the components of $\vec{v}$ to the angle of $\vec{v}$ relative to the cartesian basis vectors from the diagram

c o s α = \frac{v_{x}}{| \vec{v} |}, c o s β = \frac{v_{y}}{| \vec{v} |}, c o s γ = \frac{v_{z}}{| \vec{v} |}

cartesian coordinates -> basis vectors are fixed in direction everywhere in space
- this is not true for polar coordinates
for most of the time, right-handed cartesian coordinates will be used