Question

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?

Answer 1

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

Explanation:

You have to do a cross product to find a vector perdendicular to `veca` and `vecb`.

The cross product is given by the determinant

`| (hati,hatj,hatk), (1,13,1), (1,6,1) |`

`=hati(13-6)-hatj(1-1)+hatk(6-13)`

`=〈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)=7sqrt2`

The unit vector `=1/(7sqrt2)〈7,0,-7〉`

`=〈1/sqrt2,0,-1/sqrt2〉`