Angle between Two Vectors

The discussion on direction angles of vectors focused on finding the angle of a vector with respect to the positive x-axis. This discussion will focus on the angle between two vectors in standard position. A vector is said to be in standard position if its initial point is the origin (0, 0).

Figure 1 shows two vectors in standard position.



The angle between two vectors in standard position can be calculated as follows:

ANGLE BETWEEN TWO VECTORS:


If θ is the angle between two non-zero vectors in standard position u and v:



Where 0θ2π and v= v 1 2 + v 2 2


Let's look at some examples.

To work these examples requires the use of various vector rules. If you are not familiar with a rule go to the associated topic for a review.


Example 1: Find the angle θ between u=6,3 and v=5,13 .

Step 1: Find the dot product of the vectors.


Remember the result will be a scalar.


u·v= u 1 v 1 + u 2 v 2

u·v


( 6·5 )+( 3·13 )


30+39=69

Step 2: Find the magnitudes of each vector.


v= v 1 2 + v 2 2

||u|| = u 1 2 + u 2 2


||u|| = 6 2 + 3 2


||u|| = 45 =3 5


______________________________


||v|| = v 1 2 + v 2 2


||v|| = 5 2 + 13 2


||v|| = 194

Step 3: Substitute and solve for θ.


cos θ= u·v u·v

cos θ= u·v u·v


cos θ= 69 3 5 · 194 = 23 970


θ= cos 1 23 970


θ42°

Example 2: Find the angle θ between u=3,-6 and v=8,4 .

Step 1: Find the dot product of the vectors.


Remember the result will be a scalar.


u·v= u 1 v 1 + u 2 v 2

u·v


( 3·8 )+( 6·4 )


2424=0

Step 2: Find the magnitudes of each vector.


v= v 1 2 + v 2 2

||u|| = u 1 2 + u 2 2


||u|| = 3 2 + ( 6 ) 2


||u|| = 45 =3 5


______________________________


||v|| = v 1 2 + v 2 2


||v|| = 8 2 + 4 2


||v|| = 80 =2 20

Step 3: Substitute and solve for θ.


cos θ= u·v u·v


As soon as you determine that the dot product is 0 you do not need to calculate the magnitudes. They are completed here for your benefit.


Note that when two vectors in standard position have a dot product of 0 the angle between them is 90°.

cos θ= u·v u·v


cos θ= 0 3 5 ·2 20 = 0 6 100 = 0 60


θ= cos 1 0


θ=90°





Related Links:
Math
algebra
Decomposing a Vector into Components
The First Derivative Rule
Pre Calculus


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