# What is the cross product of (2i -3j + 4k) and (i + j -7k)?

$17 i + 18 j + 5 k$

#### Explanation:

The cross-product of vectors $\left(2 i - 3 j + 4 k\right)$ & $\left(i + j - 7 k\right)$ is given by using determinant method

$\left(2 i - 3 j + 4 k\right) \setminus \times \left(i + j - 7 k\right) = 17 i + 18 j + 5 k$

Jun 27, 2018

The vector is = 〈17,18,5〉

#### 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=〈2,-3,4〉 and vecb=〈1,1,-7〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(2 , - 3 , 4\right) , \left(1 , 1 , - 7\right) |$

$= \vec{i} | \left(- 3 , 4\right) , \left(1 , - 7\right) | - \vec{j} | \left(2 , 4\right) , \left(1 , - 7\right) | + \vec{k} | \left(2 , - 3\right) , \left(1 , 1\right) |$

$= \vec{i} \left(\left(- 3\right) \cdot \left(- 7\right) - \left(4\right) \cdot \left(1\right)\right) - \vec{j} \left(\left(2\right) \cdot \left(- 7\right) - \left(4\right) \cdot \left(1\right)\right) + \vec{k} \left(\left(2\right) \cdot \left(1\right) - \left(- 3\right) \cdot \left(1\right)\right)$

=〈17,18,5〉=vecc

Verification by doing 2 dot products

〈17,18,5〉.〈2,-3,4〉=(17)*(2)+(18)*(-3)+(5)*(4)=0

〈17,18,5〉.〈1,1,-7〉=(17)*(1)+(18)*(1)+(5)*(-7)=0

So,

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