Question

How do you find a unit vector u that is orthogonal to a and b where a = 7 i - 5 j + k and b = -7 j - 5 k?

Answer 1

Please see the explanation for steps leading to the unit vector:

`hatc = 1/(5sqrt(158))(-18hati + 35hatj - 49hatk)`

Explanation:

The cross product of the two given vectors will give you a vector that is orthogonal; to make it a unit vector, you merely divide it by its magnitude.

`barc = bara xx barb = | (hati, hatj, hatk, hati, hatj), (7,5,1,7,5), (0,-7,-5,0,-7) | =`

`barc = hati{(5)(-5) - (1)(-7)} + hatj{(1)(0) - (7)(-5)} + hatk{(7)(-7) - (5)(0)} =`

`barc = -18hati + 35hatj - 49hatk`

The above vector, `barc` is orthogonal to both `bara and barb`.

To make `barc` a unit vector, compute its magnitude and then divide the vector by it:

`|barc| = sqrt((-18)^2 + 35^2 + (-49)^2)`

`|barc| = sqrt(324 +1225 + 2401)`

`|barc| = sqrt(3950) = 5sqrt(158)`

The unit vector is:

`hatc = 1/(5sqrt(158))(-18hati + 35hatj - 49hatk)`