# Find a vector which is perpendicular to both vector A and B, where A=2i+3j+4k B=i+2j+3k?

Jul 10, 2017

The vector is =〈1,-2,1〉

#### Explanation:

The vector perpendicular to 2 vectors is calculated with the determinant (cross product)

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

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

Here, we have veca=〈2,3,4〉 and vecb=〈1,2,3〉

Therefore,

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

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

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

=〈1,-2,1〉=vecc

Verification by doing 2 dot products

〈1,-2,1〉.〈2,3,4〉=1*2-2*+1*4=0

〈1,-2,1〉.〈1,2,3〉=1*1-2*2+1*3=0

So,

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

Jul 10, 2017

$i - 2 j + k$

#### Explanation:

Cross product of vectors A and B is perpendicular to each vector A and B.

$\vec{A}$x $\vec{B} = i \left(9 - 8\right) - j \left(6 - 4\right) + k \left(4 - 3\right)$

$\vec{A}$x $\vec{B} = i - 2 j + k$

Jul 10, 2017

We know that cross product of any two vectors yields a vector which is perpendicular to both vectors
$\therefore$ for two vectors $\vec{A} \mathmr{and} \vec{B}$ if $\vec{C}$ is the vector perpendicular to both.
$\vec{C} = \vec{A} \times \vec{B} =$$\left[\begin{matrix}\hat{i} & \hat{j} & \hat{k} \\ {A}_{1} & {A}_{2} & {A}_{3} \\ {B}_{1} & {B}_{2} & {B}_{3}\end{matrix}\right]$
=(A_2B_3−B_2A_3)hati−(A_1B_3−B_1A_3)hatj+(A_1B_2−B_1A_2)hatk.
Inserting given vectors we obtain
$\vec{C} =$$\left[\begin{matrix}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 1 & 2 & 3\end{matrix}\right]$
=(3xx3-2xx4)hati−(2xx3−1xx4)hatj+(2xx2−1xx3)hatk.
=hati−2hatj+hatk.