# If a= i +6j+k and b= i+13j+k, how do you find a unit vector with positive first coordinate orthogonal to both a and b?

Dec 10, 2016

The unit vector is =〈1/sqrt2,0,-1/sqrt2〉

#### Explanation:

You have to do a cross product to find a vector perdendicular to $\vec{a}$ and $\vec{b}$.

The cross product is given by the determinant

$| \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(1 , 13 , 1\right) , \left(1 , 6 , 1\right) |$

$= \hat{i} \left(13 - 6\right) - \hat{j} \left(1 - 1\right) + \hat{k} \left(6 - 13\right)$

=〈7,0,-7〉

Verification by doing the dot products

〈7,0,-7〉.〈1,6,1〉=7-7=0

〈7,0,-7〉.〈1,13,1〉=7-7=0

The unit vector is obtained by dividing withe the modulus

The modulus $= \sqrt{49 + 49} = 7 \sqrt{2}$

The unit vector =1/(7sqrt2)〈7,0,-7〉

=〈1/sqrt2,0,-1/sqrt2〉