# What is the cross product of [3,1,-5] and [2, -1, 1] ?

May 12, 2016

$A X B = - 4 i - 13 j - 5 k$

#### Explanation:

$\vec{A} = \left[3 , 1 , - 5\right]$

$\vec{B} = \left[2 , - 1 , 1\right]$

${A}_{x} = 3$
${A}_{y} = 1$
${A}_{z} = - 5$

${B}_{x} = 2$
${B}_{y} = - 1$
${B}_{z} = 1$

$A X B = \left({A}_{y} \cdot {B}_{z} - {A}_{z} \cdot {B}_{y}\right) i - \left({A}_{x} \cdot {B}_{z} - {A}_{z} \cdot {B}_{x}\right) j + \left({A}_{x} \cdot {B}_{y} - {A}_{y} - {B}_{x}\right) k$

$A X B = i \left(1 \cdot 1 - \left(5 \cdot 1\right)\right) - j \left(3 \cdot 1 + 2 \cdot 5\right) + k \left(- 1 \cdot 3 - 2 \cdot 1\right)$

$A X B = i \left(1 - 5\right) - j \left(3 + 10\right) + k \left(- 3 - 2\right)$

$A X B = - 4 i - 13 j - 5 k$