# What is the cross product of <4 , 5 ,-9 > and <4, 3 ,0 >?

Apr 8, 2018

The vector is =〈27,-36,-8〉

#### Explanation:

The cross product of 2 vectors is calculated with the determinant

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(d , e , f\right) , \left(g , h , i\right) |$

where veca=〈d,e,f〉 and vecb=〈g,h,i〉 are the 2 vectors

Here, we have veca=〈4,5,-9〉 and vecb=〈4,3,0〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(4 , 5 , - 9\right) , \left(4 , 3 , 0\right) |$

$= \vec{i} | \left(5 , - 9\right) , \left(3 , 0\right) | - \vec{j} | \left(4 , - 9\right) , \left(4 , 0\right) | + \vec{k} | \left(4 , 5\right) , \left(4 , 3\right) |$

$= \vec{i} \left(\left(5\right) \cdot \left(0\right) - \left(- 9\right) \cdot \left(3\right)\right) - \vec{j} \left(\left(4\right) \cdot \left(0\right) - \left(4\right) \cdot \left(- 9\right)\right) + \vec{k} \left(\left(4\right) \cdot \left(3\right) - \left(4\right) \cdot \left(5\right)\right)$

=〈27,-36,-8〉=vecc

Verification by doing 2 dot products

〈27,-36,-8〉.〈4,5,-9〉=(27)*(4)+(-36)*(5)+(-8)*(-9)=0

〈27,-36,-8〉.〈4,3,0〉=(27)*(4)+(-36)*(3)+(-8)*(0)=0

So,

$\vec{c}$ is perpendicular to $\vec{a}$ and $\vec{b}$

Apr 8, 2018

$< 27 , - 36 , - 8 >$

#### Explanation:

The easiest way to compute is to make a matrix whose determinant is the cross product.

M = [ (hat i, hat j , hat k), (v_(1x), v_(1y), v_(1z)), (v_(2x), v_(2y), v_(2z)) ]

For our vectors we have:

$\implies {v}_{1} = < 4 , 5 , - 9 >$
$\implies {v}_{2} = < 4 , 3 , 0 >$

Hence, we have:

M =[ (hat i, hat j , hat k), (4, 5, -9), (4, 3, 0) ]

Now we just need to take the determinant of this matrix to get the cross product.

${v}_{1} \otimes {v}_{2} = \text{det} \left(M\right)$

$= \left(5 \cdot 0 - \left(- 9\right) \cdot 3\right) \hat{i} + \left(\left(- 9\right) \cdot 4 - \left(4\right) \cdot 0\right) \hat{j} + \left(4 \cdot 3 - \left(5\right) \cdot 4\right) \hat{k}$

$= \left(27\right) \hat{i} + \left(- 36\right) \hat{j} + \left(- 8\right) \hat{k}$

$= 27 \hat{i} - 36 \hat{j} - 8 \hat{k}$

Hence:

${v}_{1} \otimes {v}_{2} = < 27 , - 36 , - 8 >$