Direction Angles of Vectors



Figure 1 shows a unit vector u that makes an angle θ with the positive x-axis. The angle θ is called the directional angle of vector u.

The terminal point of vector u lies on a unit circle and thus u can be denoted by:

u=x,y=cosθ,sinθ=(cosθ)i+(sinθ) j



Any vector that makes an angle θ with the positive x-axis can be written as the unit vector times the magnitude of the vector.

v=v(cosθ)i+v(sinθ) j

v=ai+bj



Therefore the direction angle of θ of any vector can be calculated as follows:

DIRECTIONAL ANGLE:

tan θ= sin θ cosθ = vsin θ vcosθ = b a



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 direction angle of w = -2i + 9j.

Step 1: Identify the values for a and b and calculate θ.


tan θ= b a

a = -2, b = 9


tanθ= b a = 9 2


θ= tan 1 | 9 2 |


θ78°

Step 2: Determine the Quadrant the vector lies in.

Because the vector terminus is (-2, 9), it will fall in quadrant II and so will θ.

Step 3: Make any necessary adjustments to find the directional angle θ from the positive x-axis.

Since the reference angle is 78°, the directional angle from the positive x-axis is 180° - 78° = 102°.

Example 2:Find the direction angle of v=3( cos60°i+sin60°j ) .

Step 1: Simplify vector v using scalar multiplication.


kv=k v 1 , v 2 =k v 1 ,k v 2 Scalar Multiplication

v=3( cos60°i+sin60°j )


v=3·cos60°i+3 ·sin60°j


v=3· 1 2 i+3 · 3 2 j


v= 3 2 i+ 3 3 2

Step 2: Identify the values for a and b and calculate θ.

a= 3 2 , b= 3 3 2


tanθ= b a = 3 3 2 3 2 = 3 3 2  · 2 3 = 3


θ= tan 1 | 3 |


θ=60°

Step 3: Determine the Quadrant of the vector lies in.

Because the vector terminus is ( 3 2 ,  3 3 2 )=( 1.5,2.6 ) and both components are positive the vector will fall in quadrant I and so will θ.

Step 4: Make any necessary adjustments to find the directional angle θ from the positive x-axis.

Since the reference angle is 60°, the directional angle from the positive x-axis is 60° - 0° = 60°.





Related Links:
Math
algebra
The Dot Product
Angle between Two Vectors
Pre Calculus


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