- #1
Mayhem
- 353
- 251
...and is it ever useful?
An arbitrary complex number has the form ##z = a + bi## where ##a, b \in \mathbb{R}## and the dot product of two arbitrary vectors ##\vec{v} = \binom{v_1}{v_2}## and equivalently for vector ##\vec{w}## is ##\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_3## Then the ##z## may be expressed as ##z = a+bi = \vec{r} \cdot \vec{c} = \binom{a}{b} \cdot \binom{1}{i}## where ##\vec{r}## denotes the real part and ##\vec{c}## the imaginary part.
Then in the complex plane, their composite vector ##\vec{p}## may be expressed using their magnitudes, giving ##\vec{p} = \binom{||r||}{||c||} = \binom{\sqrt{a^2+b^2}}{\sqrt{2}i}##
Math seems valid unless I made a stupid mistake. Is this ever useful?
An arbitrary complex number has the form ##z = a + bi## where ##a, b \in \mathbb{R}## and the dot product of two arbitrary vectors ##\vec{v} = \binom{v_1}{v_2}## and equivalently for vector ##\vec{w}## is ##\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_3## Then the ##z## may be expressed as ##z = a+bi = \vec{r} \cdot \vec{c} = \binom{a}{b} \cdot \binom{1}{i}## where ##\vec{r}## denotes the real part and ##\vec{c}## the imaginary part.
Then in the complex plane, their composite vector ##\vec{p}## may be expressed using their magnitudes, giving ##\vec{p} = \binom{||r||}{||c||} = \binom{\sqrt{a^2+b^2}}{\sqrt{2}i}##
Math seems valid unless I made a stupid mistake. Is this ever useful?