unit vector othogonal to two vectors?

Question

find a unit vector orthogonal to both vectors
(1, -3, 2) and (-1, 2, 3).

are there any other ways besides using the cross product?

Answer 1

The cross product is the easiest, but you could look for a vector whose dot product with both vectors is zero. Since the cross product only exists in three dimensions, you couldn't use it anyway in circumstances involving more or less dimensions than three.

Let
n = be normal to the vectors
<1,-3,2> and
<-1,2,3>

Take the dot product of n with both vectors and set it equal to zero. Solve for a,b, and c.

•<1,-3,2> = a - 3b + 2c = 0
•<-1,2,3> = -a + 2b + 3c = 0

So we have

a - 3b + 2c = 0
-a + 2b + 3c = 0

Adding the two equations we get

-b + 5c = 0
b = 5c

Plug this back into the first equation.

a - 3b + 2c = 0
a - 3(5c) + 2c = 0
a - 15c + 2c = 0
a - 13c = 0
a = 13c

We now have a normal vector n.

n = <13c,5c,c>
We can choose any non-zero value for c. Let c = 1.

n = <13,5,1>

Test its orthoganality.

<13,5,1>•<1,-3,2> = 13 - 15 + 2 = 0
<13,5,1>•<-1,2,3> = -13 + 10 + 3 = 0

So it is orthogonal. To make it a unit vector, divide it by its magnitude.

|| n || = √(13² + 5² + 1²) = √(169 + 25 + 1) = √195

So the desired unit normal vector is:

<13,5,1> / √195

Answer 2

Hi,

You want to use the cross product. For your vectors, (1, -3, 2) and (-1, 2, 3), the cross product would be:

(1, -3, 2)
(-1, 2, 3) = (-3)(3)-(2)(2), (2)(-1)-(1)(3), (1)(2)-(-3)(-1) =

-9 - 4, -2 - 3, 2 - 3 = (-13, -5, -1)

This vector is orthogonal, but it is not a unit vector. To make the unit vector, you need to find its magnitude from the distance formula.

magnitude = sqrt((-13)^2 + (-5)^2 + (-1)^2) =
sqrt( 169 + 25 + 1) = sqrt(195) = 13.96

Now divide each number in your perpendicular vector by this and that gives you the coordinates of your orthogonal unit vector as

(-13/sqrt(195), -5/sqrt(195), -1/sqrt(105)) or

(-.9213, -.3582, -.0716) are 2 forms of the answer.

I hope that helps you.

Answer 3

You could definitely calculate the unit vector orthogonal to both by writing a set of trig equations, but it gets to be a big pain.

The cross product is the easiest way to do this.

Answer 4

Find a vector that is orthogonal to both vectors, find its norm and divide it by the norm

Ana

Answer 5

Probably... but the cross product is the easiest way I've seen.