I assume by * you mean X (vector product). So yes, ##e_x \times e_y = e_z##. What about ##e_y \times e_x##, ##e_x \times e_z##, and the other combinations? (You are making a sign error.)Roodles01 said:now, these are all orthogonal to each other, so, for example, if I have ex * ey then I should end up with ez, shouldn't I?
Vector multiplication is a mathematical operation that combines two or more vectors to produce a new vector. It is used to calculate the direction and magnitude of a resultant vector, which represents the combined effect of the individual vectors.
There are two types of vector multiplication: dot (or scalar) product and cross (or vector) product. The dot product results in a scalar quantity, while the cross product results in a vector quantity.
The dot product of two vectors is calculated by multiplying the corresponding components of the vectors and then adding the products. For example, the dot product of vectors a and b would be written as a · b and calculated as axbx + ayby + azbz, where ax, ay, and az are the x, y, and z components of vector a, and bx, by, and bz are the x, y, and z components of vector b.
The cross product of two vectors is calculated by taking the determinant of a 3x3 matrix. The resulting vector is perpendicular to both of the original vectors and its direction can be determined using the right-hand rule. The magnitude of the cross product is equal to the product of the magnitudes of the original vectors multiplied by the sine of the angle between them.
The dot product of two vectors can be interpreted as the projection of one vector onto the other, multiplied by the magnitude of the second vector. The cross product can be interpreted as the area of the parallelogram formed by the two vectors, multiplied by a unit vector perpendicular to the plane of the parallelogram. These interpretations can be used to better understand the direction and magnitude of the resultant vector.