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KingNothing
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Say I'm given a random 3-dimensional vector, pointing from the origin. How can I find the angle it forms with a plane defined by two other vectors?
A 3D vector is a mathematical representation of a direction and magnitude in three-dimensional space. It is typically denoted by three coordinates (x, y, z) and can be used to describe the position, velocity, or acceleration of an object in three-dimensional space.
A plane is a flat, two-dimensional surface that extends infinitely in all directions. In three-dimensional space, a plane can be defined by a point and a normal vector, which is perpendicular to the plane.
To find the angle between a 3D vector and a plane, you can use the dot product formula: θ = cos^-1((v · n) / (|v| * |n|)), where v is the 3D vector and n is the normal vector of the plane. This formula gives you the angle in radians, so you may need to convert it to degrees if necessary.
Yes, the angle between a 3D vector and a plane can be negative. The dot product formula gives the signed angle between the two, meaning it takes into account the direction of the vectors. If the angle is negative, it means the vector is pointing away from the plane, and if it is positive, it is pointing towards the plane.
Knowing the angle between a 3D vector and a plane is useful in many scientific fields, such as physics, engineering, and computer graphics. It can help determine the orientation of an object in space, calculate the angle of incidence and reflection for light or sound waves, and create 3D models and simulations.