I'm trying to get the conjugate of a Quaternion but I'm not sure if I'm using the formula correctly.
I have a Quaternion of: $q = 2 - i + j + 3k$
and with help online, the conjugate of a Quaternion is defined by:
$a^* = a_1 - a_2 i - a_3 j - a_4 k$
But if I apply this to the Quaternion I have it looks as if it is just subtracting each ijk from each other, or maybe I'm just not understanding this correctly.
I'd like to better understand what is going on here and how to get the conjugate of a Quaternion properly. I'm happy to follow any web links that better explain the method of doing this.
Thank you in advance.
A quaternion $q$ can be written in one and only one way as $$ q=a_0+a_1i+a_2j+a_3k $$ with real $a_0$, $a_1$, $a_2$ and $a_3$. By definition, the conjugate of $q$ is $$ q^*=a_0+(-a_1)i+(-a_2)j+(-a_3)k $$ (but usually written as $q^*=a_0-a_1i-a_2j-a_3k$). In your case $$ a_0=2,\quad a_1=-1,\quad a_2=1,\quad a_3=3 $$ so $$ -a_1=1,\quad -a_2=-1,\quad -a_3=-3 $$ and therefore $$ q^*=2+1i+(-1)j+(-3)k $$ usually written, for short, $q^*=2+i-j-3k$.