# Unit vector perpendicular to two known vectors

1. Jan 10, 2010

### cjwalle

1. The problem statement, all variables and given/known data
Find a unit vector which is perpendicular to both of the vectors a = 4i + 2j - 3k and b = 2i - 3j + k

c = xi + yj + zk

2. Relevant equations
a$$\bot$$c $$\longrightarrow$$ a $$\bullet$$ c

3. The attempt at a solution
Okay, here's what I've done so far.

Take the dot-product of a and c, and b and c
a $$\bullet$$b: 4x + 2y -3z = 0
b $$\bullet$$b: 2x - 3y + z = 0

(1) 4x + 2y -3z = 0
(2) 2x - 3y + z = 0

I isolate z and get rid of x by multiplying (1) with -2 and (2) with 4, then add them:
(1) -8x -4y = -6z
+
(2) 8x -12y = 5z

-16y = 10z

y/z = 10/16

Which again means that:
y = 10m
z = 16m
where m is a constant and $$\neq$$ 0

And then I insert this into (1) to find x:
4x + 2(10m) - 3(16m) = 0
4x + 20m - 48m = 0
x = 7m

c = m(7i + 10j + 16k)

For the easiest possible solution, m = 1.
c = 7i + 10j + 16k
As far as I can tell, this is a perfectly valid answer.

(1/9$$\sqrt{5}$$)(7i + 10j + 16k)

While this does not contradict my solution, that is a far too weird m to have been chosen randomly. Does anyone see how they were thinking?

Thanks
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 10, 2010

### Fightfish

The question asked for a unit vector ie a vector with magnitude of 1. Hence m must be chosen so that the magnitude of the vector is one.
Also, there is a much much easier way of solving the problem. The cross product of two vectors yields a third vector that is perpendicular to both original vectors. In this case, simply take the cross product with a with b, and adjust to obtain a unit vector.

3. Jan 10, 2010

### cjwalle

*facepalm*

Yes, of course.

And this exercise is from the sub-chapter before cross products, so I figured I'd try to do it the way they wanted me to do it.

Thanks a lot :)

Edit: Just to complete the solution:

| c | = $$\sqrt{7^{2} + 10^{2} + 16^{2}}$$ = $$\sqrt{405}$$
= $$\sqrt{81}$$$$\sqrt{5}$$ = 9$$\sqrt{5}$$

Thus m = 1/ 9$$\sqrt{5}$$

Last edited: Jan 10, 2010