Calculating the angle between two vectors

In summary: This can be done by choosing a vector z that is perpendicular to the plane where the two vectors u and v lie. Then, you can use the inverse cosine function to calculate the angle between u and v, and adjust it to fit within the range of 0 to 2pi by considering the orientation of z.
  • #1
computerex
68
0
When calculating the angle between two vectors, I take the inverse cosine of the dot product of the two vectors normalized. This gives me the angle, however this does not specify whether the direction is backwards or forwards. In other words, it'll give me a range of -PI/2 to PI/2, then it loops back. Is there a way to figure out the angle in range of 0 to 360 degrees with this?
 
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  • #2
Usually (for how it is normally defined, for example, in calculators) the cos^(-1) gives back values belonging to [0, pi], not [-pi/2 , pi/2] because in this interval the cos is not invertible.

Lets consider R^3.
If you want an angle between two normalized vectors u and v, belonging to [0, 2pi[, you have to define an orientation z for the plane where the two vectors are (the vector z lies in one side or the other of the plane, you choose).

(1) calculate alpha = cos^(-1) (u.v)
(2) calculate w = u x v
(3) if w.z > 0 or w.z = 0 then the answer is alpha
(4) if w.z < 0 then the answer is 2pi - alpha
 
  • #3
The "angle between two vectors" is, by definition, the smaller of the two angles formed by the intersection of lines in the direction of the two vectors. The angle between two vectors is always between 0 and [itex]\pi[/itex].
 
  • #4
HallsofIvy said:
The "angle between two vectors" is, by definition, the smaller of the two angles formed by the intersection of lines in the direction of the two vectors. The angle between two vectors is always between 0 and [itex]\pi[/itex].

Nevertheless, if you need it, you can define the angle in (0, 2pi), and to do so you have to decide an orientation.
 
  • #5


Yes, there is a way to calculate the angle between two vectors in the range of 0 to 360 degrees. One approach is to use the cross product of the two vectors to determine the direction of rotation. The cross product will give a vector perpendicular to both input vectors, and its direction can be used to determine whether the angle is positive or negative. By combining the inverse cosine of the dot product with the direction from the cross product, you can calculate the angle in the range of 0 to 360 degrees. Another approach is to use the atan2 function, which takes into account both the x and y components of the vectors to determine the angle in the range of 0 to 360 degrees. Both of these methods can be used to calculate the angle between two vectors with a specified direction.
 

FAQ: Calculating the angle between two vectors

How do you calculate the angle between two vectors?

To calculate the angle between two vectors, you can use the dot product formula: θ = cos^-1 (a · b / |a||b|), where a and b are the two vectors and |a| and |b| represent their magnitudes. This will give you the angle in radians. Alternatively, you can use the cross product formula: θ = sin^-1 (|a x b| / |a||b|), where a x b is the cross product of the two vectors and |a| and |b| represent their magnitudes. This will give you the angle in degrees.

What is the dot product of two vectors?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and returns a scalar quantity. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. The dot product is used to find the angle between two vectors, and it also has applications in physics and engineering.

How do you find the magnitude of a vector?

The magnitude of a vector is the length or size of the vector. To find the magnitude, you can use the Pythagorean theorem: |v| = √(vx^2 + vy^2 + vz^2), where vx, vy, and vz are the components of the vector in the x, y, and z directions, respectively. Alternatively, you can use the distance formula: |v| = √(x^2 + y^2 + z^2), where x, y, and z are the coordinates of the endpoint of the vector.

Can the angle between two vectors be negative?

No, the angle between two vectors cannot be negative. The angle is always measured as the shortest rotation from one vector to the other in a counterclockwise direction. If the vectors are pointing in opposite directions, the angle will be 180 degrees or π radians.

What is the difference between a scalar and a vector quantity?

A scalar quantity is a physical quantity that has only magnitude, such as temperature, mass, or speed. A vector quantity is a physical quantity that has both magnitude and direction, such as velocity, force, or displacement. The angle between two vectors is an example of a vector quantity, as it has both magnitude (the angle) and direction (the direction of rotation).

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