Question 862821: Determine which of the given vectors are orthogonal:
CHoose from the following answers.
a) u=7i +7j, v=7j+7i
b)u=7j v=7i
c)u=7i, v=7j
d)u=7i, v=7i
I cannot recall the book going over this, in this form. Please help if you can. I need help understanding this. Thank you
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! note that u*v is defined as the dot product and the result of taking the dot product is a scaler
u*v = u1v1 + u2v2
a) u*v = 7*7 + 7*7 = 49 + 49 = 98
b) u*v = 0*7 + 7*0 = 0 + 0 = 0
c) u*v = 7*0 + 0*7 = 0 + 0 = 0
d) u*v = 7*7 + 0*0 = 49 + 0 = 49
b and c are orthogonal
here is the math behind this
one of the properties of dot product is v*v = ||v||^2, this is the magnitude of a vector
use law of cosines to determine the length of the distance between u and v
d(u,v)^2 = ||u||^2 + ||v||^2 - 2 ||u|| ||v|| cos(t)
using definition of distance and the definition above of magnitude of a vector
(v1 - u1)^2 + (v2 - u2)^2 = (v1^2 + v2^2) + (u1^2 + u2^2) - 2 ||u|| ||v|| cos(t)
expanding the squares on the left of the =, and simpfying, we get
v1u1 + v2u2 = ||u|| ||v|| cos(t)
the left side of the = is just the dot product of vectors u and v
u*v = ||u|| ||v|| cos(t)
cost(t) = u*v / ||u|| ||v||
NOTE that cos(t) = 0 ( t = 90 degrees) the dot product u * v = 0 leads us to
vectors v and u are orthogonal if and only if v * u = 0
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