Go
to the 
Go
to the
You can use the geometric definition of the dot
product to calculate the angle between two non-zero vectors. Now,
if one of the vectors is the zero vector, the angle between the two vectors
is not defined at all. For two non-zero vectors u and v,
solve the formula
u•v = ||u|| ||v|| cos θ for cos θ:
u•v = ||u|| ||v|| cos θ for cos θ:
.
Then
.
Some special cases of this formula are useful to remember.
Two vectors are perpendicular, or orthogonal, when the angle between them is 90º or π/2. In that case, cos θ = 0 and you get u•v = 0.
If the angle between the vectors is acute (less than π/2), then cos θ >
0, so
u•v > 0.
If the angle between the vectors is obtuse (greater than π/2), then cos θ <
0, so
u•v < 0.
If the vectors are parallel in the same direction, then the angle between
them is 0 and
cos θ = 1 , so u•v = ||u||
||v||.
If the vectors are parallel in opposite directions, then the angle between
them is π and
cos θ = -1, so u•v =
-||u||
||v||.
 Dot
Products of Vectors |
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| Introduction | Two definitions of dot products | Calculation rules for dot products | Finding angles with dot products | Orthogonal projections |