Dot Product For An Angle
Three points and in three-dimensional Euclidean space have their respective coordinates and What is the measure of
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Taking the direction from origin to the points A, B and C, we can write them as position vectors: O A = − 6 i ^ + 2 j ^ − 4 k ^ O B = − i ^ + j ^ − 2 k ^ O C = − 2 i ^ + 2 j ^ + k ^ .
Now lets figure out the vectors B A , B C , which is: B A = O A − O B = − 5 i ^ + j ^ − 2 k ^ B C = O C − O B = − i ^ + j ^ + 3 k ^ .
Since the dot product of the vectors B A , B C is B A ⋅ B C = ( − 5 ) ( − 1 ) + ( 1 ) ( 1 ) + ( − 2 ) ( 3 ) = 0 , the vectors B A , B C are perpendicular.
Thus ∠ A B C = 9 0 ∘ . □