- #1
htk
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can anyone please help me how to find the direction angle of a vector? Thank you!
In summary, to find the direction angle of a two-dimensional vector, you can use the formula \tan \theta = b / a, but you also need to consider in which quadrant the vector lies and adjust the angle accordingly.
- #2
Dragonfall
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Think about the triangle that the vector and its components form.
- #3
HallsofIvy
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I presume you are talking about vectors in the plane since vectors in three dimensions have three "direction angles". How are you given the vector? If in x,y components, say ai+ bj, then b/a is the tangent of the angle the vector makes with the x-axis:htk said:
[tex]\theta= arctan(\frac{a}{b})[/tex].
- #4
tiny-tim
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Welcome to PF!
Hi htk! Welcome to PF!
You find the cosine of the angle …
Hi htk! Welcome to PF!
htk said:
You find the cosine of the angle …
which you do by finding the dot product.
- #5
HallsofIvy
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His question was about a single vector. What do you want him to take the dot product with?tiny-tim said:Hi htk! Welcome to PF!
You find the cosine of the angle …
which you do by finding the dot product.
- #6
tiny-tim
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… just answering the question as asked! …
with whatever mysterious entity he had in mind when he specified …
HallsofIvy said:
with whatever mysterious entity he had in mind when he specified
…htk said:
- #7
Elucidus
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A two-dimensional vector [itex]\langle a,b \rangle[/itex] will have a direction angle [itex]\theta \text{ such that } \tan \theta = b / a[/itex] (not (a/b)) but this does not uniquely determine [itex]\theta[/itex], even if it is restricted to the interval [itex][0, 2 \pi )[/itex].
You also need to consider in which quadrant does the vector lie. You need to adjust the value of [itex]\theta[/itex] so that it falls into the correct quadrant.
For example, the vector [itex]\langle -3, 3 \rangle[/itex] has a direction angle so that [itex]\tan \theta = 3 / -3 = -1 \text{ which implies } \theta = -\pi /4 + n \pi[/itex] for an appropriate choince of integer n. Since the vector is in the second quadrant, we need to select the angle to fall there, so [itex]\theta = 3\pi / 4[/itex] (here n = 1).
I hope this helps.
--Elucidus
You also need to consider in which quadrant does the vector lie. You need to adjust the value of [itex]\theta[/itex] so that it falls into the correct quadrant.
For example, the vector [itex]\langle -3, 3 \rangle[/itex] has a direction angle so that [itex]\tan \theta = 3 / -3 = -1 \text{ which implies } \theta = -\pi /4 + n \pi[/itex] for an appropriate choince of integer n. Since the vector is in the second quadrant, we need to select the angle to fall there, so [itex]\theta = 3\pi / 4[/itex] (here n = 1).
I hope this helps.
--Elucidus
- #8
Dragonfall
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HallsofIvy said:
I suppose he could dot it with (1,0).
EDIT: Modulo sign.
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FAQ: Find Direction Angle of Vector in Plane
What is the concept of finding the direction angle of a vector in a plane?
The direction angle of a vector in a plane refers to the angle that the vector makes with the positive x-axis in a two-dimensional coordinate system. It is used to describe the orientation of a vector in relation to the horizontal axis.
How do you calculate the direction angle of a vector in a plane?
The direction angle of a vector can be calculated using trigonometric functions. First, find the components of the vector in the x and y directions. Then, use the formula θ = tan-1(y/x) to calculate the direction angle.
What is the range of values for the direction angle of a vector in a plane?
The direction angle of a vector can have a range of values from 0° to 360°, or from 0 to 2π radians. This range covers all possible orientations of a vector in a two-dimensional plane.
Can the direction angle of a vector in a plane be negative?
Yes, the direction angle of a vector can be negative if the vector is in the third or fourth quadrant of the coordinate plane. In this case, the angle is measured clockwise from the positive x-axis, resulting in a negative value.
How is the direction angle of a vector in a plane used in real-world applications?
The concept of direction angle of a vector is used in various fields such as physics, engineering, and navigation. It helps in determining the direction of force or motion, analyzing the orientation of objects, and calculating the bearing or heading in navigation systems.