Have you read this: https://en.wikipedia.org/wiki/Cross_product#Geometric_meaning ?prashant singh said:What does the angle theta acutally means in cross product because I have seen in many places it is written that theta is the angle at which two vector on a given plane will coinside with each other so that there will be only one direction. Is it true and why they defined it in this way , I need more information on this
fresh_42 said:Have you read this: https://en.wikipedia.org/wiki/Cross_product#Geometric_meaning ?
A vector is a direction and a length. Where you "connect" them depends on what you want to do.prashant singh said:If there are two vector in a given plane and they are not connected tail to tail then what we have to doo. I think now we have to connect the two vectors tail to tail by the smallest angle theta so that we can have a vector perpendicular to both the vectors. Correct me if I am wrong
robphy said:In the cross-product, the first goal should be to determine the signed-area of the parallelogram formed by the vectors.
Slide the vectors to have their tails meet, then measure the angle from the first vector to the second vector.
Then you'll take the sine of that angle.
In three dimensions, if this area is nonzero,
there will be a unique line that is perpendicular to that parallelogram.
The cross product, also known as the vector product, is a mathematical operation that combines two vectors to produce a new vector that is perpendicular to both of the original vectors. It is represented by the symbol "×" and is used in various fields of science, including physics, engineering, and computer graphics.
The cross product is calculated by taking the determinant of a 3x3 matrix formed by the two original vectors and the unit vectors of the coordinate system. The resulting vector is equal to the magnitude of the two vectors multiplied by the sine of the angle between them, and its direction is determined by the right-hand rule.
The cross product and dot product are two different types of vector operations. While the cross product results in a new vector, the dot product results in a scalar value. Additionally, the cross product is only applicable to 3-dimensional vectors, while the dot product can be calculated for any number of dimensions.
The cross product has many applications in science, including calculating torque in physics, determining magnetic field strength in electromagnetism, and computing normal vectors in 3D graphics. It is also used in vector calculus and differential geometry to solve various problems.
No, the cross product can only be calculated for two vectors that lie in the same plane. If the two vectors are non-planar, their cross product will be a zero vector. In such cases, the dot product can be used to find the angle between the two vectors instead.