SOLUTION: Hi all, can anyone help me with the following?; Find a unit vector in the direction of (3,1,2) and one in the opposite direction. How do you do this? Thanks, -nick

Algebra ->  Vectors -> SOLUTION: Hi all, can anyone help me with the following?; Find a unit vector in the direction of (3,1,2) and one in the opposite direction. How do you do this? Thanks, -nick      Log On


   

Question 206228: Hi all, can anyone help me with the following?;
Find a unit vector in the direction of (3,1,2) and one in the opposite direction. How do you do this?
Thanks, -nick
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
to get the unit vector you need to get the magnitude of the vector and then divide each measure used by the magnitude.
-----
in your problem, you have (i,j,k) = (3,1,2)
magnitude = sqrt%28i%5E2+%2B+j%5E2+%2B+k%5E2%29 = sqrt%283%5E2+%2B+1%5E2+%2B+2%5E2%29 = sqrt%289%2B1%2B4%29 = sqrt%2814%29.
-----
you take each measure and divide by the magnitude to get (3%2Fsqrt%2814%29,1%2Fsqrt%2814%29,2%2Fsqrt%2814%29)
-----
you simplify the fraction by multiplying each by sqrt%2814%29%2Fsqrt%2814%29 to get (3%2Asqrt%2814%29%2F14,sqrt%2814%29%2F14,2%2Asqrt%2814%29%2F14%29) which can be simplified further to get (3%2Asqrt%2814%29%2F14,sqrt%2814%29%2F14,%28sqrt%2814%29%2F7%29) although I think I would want to keep all denominators the same.
-----
your answer will either be (3%2Asqrt%2814%29%2F14,sqrt%2814%29%2F14,2%2Asqrt%2814%29%2F14%29) or (3%2Asqrt%2814%29%2F14,sqrt%2814%29%2F14,sqrt%2814%29%2F7%29)
-----
this is a unit vector because the magnitude is now 1 as shown by the following equation:
magnitude of vector =
-----
this becomes which becomes sqrt%28%28%289%2A14%29+%2B+%2814%29+%2B+%284%2A14%29%29%2F%2814%5E2%29%29 which becomes sqrt%28%2814%5E2%29%2F%2814%5E2%29%29 which becomes 1.
-----
A unit vector in the opposite direction would have the same values only negatives of whatever is there.
In this case, your unit vector in the opposite direction should be (-%283%2Asqrt%2814%29%2F14%29,-%28sqrt%2814%29%2F14%29,-%282%2Asqrt%2814%29%2F14%29%29)
-----
Note that keeping the denominator the same seemed like a good idea especially when verifying that the magnitude is 1.
-----