Need help with vectors.?

Question

Let a=3i+2j+2k ;

and b= -i+j+k

Find a unit vector that's perpendicular to both of them

Answer 1

The vector cross product is perpendicular to both. Set up a 3 by 3 matrix with [i j k] in the top row, [3 2 2] in the middle row, and [-1 1 1] in the bottom row. Evaluate, then normalize it, and you have your answer.

i (2-2) - j(3+2) + k(3+2) = -5j + 5k

The unit vector will be c = <-j/sqrt(2) + k/sqrt(2)>

Check with dot products:

a dot c = 0 - 2/sqrt(2) + 2/sqrt(2) = 0

b dot c = 0 - 1/sqrt(2) + 1/sqrt(2) = 0

That's it; c is a unit vector perpendicular to both a and b.

Answer 2

The cross product of two vectors is perpendicular to the original vectors.

You then find the magnitude of the new vector, and divide the new vector by scalar value of its magnitude to get the unit vector.

What I get for the cross product is <0,-5,5>

Does that make sens?, it should give a dot product of zero with each of the original vectors.

<3,2,2>*<0,-5,5> = 0 -10 + 10 = 0
<-1 ,1 ,1>*<0,-5,5> = 0

So what is the magnitude of <0,-5,5> ? maybe 5*sqrt(2).

Please check my work for errors.

Answer 3

Probably the easiest way is to take their cross product and normalize it. In 3-space, the cross product can be easily remembered as the determinant of a matrix as follows:

|i j k |
|a1 a2 a3|
|b1 b2 b3|

(Edit: Oh sh|t!!! It won't let me line up the columns properly. I hope you get the picture)

where the a1, a2, a3 and b1, b2, b3 are the i, j, k components of the vectors a and b. Expand by cofactors to get

i(a2b3-b2a3) - j(a1b3-b1a3) + k(a1b2-b1a2)

Now divide each component by
√((a2b3-b2a3)² + (a1b3-b1a3)² + (a1b2-b1a2)²)
(also called the 'Euclidean Norm') to get your unit vector that is perpendicular to the plane formed by a and b.

And remember that there are *2* of them (pointing in opposite directions ☺) so if aXb = c then -c is also a solution.

It's a bit of number crunching, but *I* had to do it with a slide rule (how's *that* for giving away my age ☺)


Doug

Answer 4

Perform the cross product of a x b
Find the magnitude of (a x b)

Divide the answer with the magnitude to obtain unit vector.

Answer 5

just do pythagorean theorem!!! a^2 +b^2 = c^2
c is your unit vector.