Question

How do you find a unit vector that is orthogonal to a and b where `a = −7 i + 6 j − 8 k` and `b = −5 i + 6 j − 8 k`?

Answer 1

The answer is `=1/5〈0,-4,-3〉`

Explanation:

To find a vector orthogonal to 2 other vectors, we must do a cross product.

The cross product of 2 vectors, `veca=〈a,b,c〉` and `vecb=〈d,e,f〉`

is given by the determinant

`| (hati,hatj,hatk), (a,b,c), (d,e,f) |`

`= hati| (b,c), (e,f) | - hatj| (a,c), (d,f) |+hatk | (a,b), (d,e) |`

and `| (a,b), (c,d) |=ad-bc`

Here, the 2 vectors are `veca=〈-7,6,-8〉` and `〈-5,6,-8〉`

And the cross product is

`| (hati,hatj,hatk), (-7,6,-8), (-5,6,-8) |`

`=hati| (6,-8), (6,-8) | - hatj| (-7,-8), (-5,-8) |+hatk | (-7,6), (-5,6) |`

`=hati(-48+48)-hati(56-40)+hatk(-42+30)`

`=〈0,-16,-12〉`

Verification, by doing the dot product

`〈0,-16,-12〉.〈-7,6,-8〉=0-96+96=0`

`〈0,-16,-12〉.〈-5,6,-8〉=0-96+96=0`

Therefore, the vector is perpendicular to the other 2 vectors

The unit vector isobtained by dividing by the modulus.

The modulus is `∥〈0,-16,-12〉∥=sqrt(0+16^2+12^2)=sqrt400=20`

The unit vector is `=1/20〈0,-16,-12〉=1/5〈0,-4,-3〉`