 # Find the cross product a X b and verify that it is orthogonal to both a and b. Brennan Flores 2021-09-25 Answered

Find the cross product $a×b$ and verify that it is orthogonal to both a and b.
$a=\left(2,3,0\right),b=\left(1,0,5\right)$

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Solution:
To find the cross product of two vectors a and b, where the components of both vectors are as follows
$a=<{a}_{1},{a}_{2},{a}_{3}>$
$b=<{b}_{1},{b}_{2}{,}_{3}>$
We use the following formula,
$c=a×b$
$=|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ {a}_{1}& {a}_{2}& {a}_{3}\\ {b}_{1}& {b}_{2}& {b}_{3}\end{array}|$
$=\stackrel{^}{i}|\begin{array}{cc}{a}_{2}& {a}_{3}\\ {b}_{2}& {b}_{3}\end{array}|-\stackrel{^}{j}|\begin{array}{cc}{a}_{1}& {a}_{3}\\ {b}_{1}& {b}_{3}\end{array}|+\stackrel{^}{k}|\begin{array}{cc}{a}_{1}& {a}_{2}\\ {b}_{1}& {b}_{2}\end{array}|$
$=\stackrel{^}{i}\left({a}_{2}{b}_{3}-{a}_{3}{b}_{2}\right)-\stackrel{^}{j}\left({a}_{1}{b}_{3}-{a}_{3}{b}_{1}\right)+\stackrel{^}{k}\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}\right)$
In order to prove that this vector is orthogonal to both vectors a and b, we need to show that the dot product of vector c to each vector is zero, i.e we need to show that
$c\cdot a=0$

Calculations:
We use equation, in order to find vector c which is the cross product of the two vectors a and b as follows
$c=a×b$
$=|\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ 2& 3& 0\\ 1& 0& 5\end{array}|$
$=\stackrel{^}{i}|\begin{array}{cc}3& 0\\ 0& 5\end{array}|-\stackrel{^}{j}|\begin{array}{cc}2& 0\\ 1& 5\end{array}|+\stackrel{^}{k}|\begin{array}{cc}2& 3\\ 1& 0\end{array}|$
$=\stackrel{^}{i}\left(\left(3\right)\left(5\right)-\left(0\right)\left(0\right)\right)-\stackrel{^}{j}\left(\left(2\right)\left(5\right)-\left(0\right)\left(1\right)\right)+\stackrel{^}{k}\left(\left(2\right)\left(0\right)-\left(3\right)\left(1\right)\right)$
$=15\stackrel{^}{i}-10\stackrel{^}{j}-3\stackrel{^}{k}$
Thus, the cross product of the two vector is the following vector
$c=<15,-10,-3>$
We find the dot product of vector c with vector a, and then b in order to find out whether if its orthogonal to both or now
$c\cdot a=<15,-10,-3>\cdot <2,3,0>$
$=30-30+0$
$=0$ (1)
And, the dot product of the vector c and b