-EquinoX- said:ok is this correct?
[tex]|a\times b|^2 = |a|^2|b|^2\sin^2\theta = |a|^2|b|^2(1 - \cos^2\theta)[/tex]
[tex] = |a|^2|b|^2 - |a|^2|b|^2\cos^2\theta = |a|^2|b|^2 - (a\cdot b)^2 [/tex]
-EquinoX- said:well yes, but what's meant here by ||v|| here is the magnitude of vector v right? now I am confused though, by definition of the geometric, it says that:
a x b = ||a|| ||b|| sin x
and it's not
|a x b| = ||a|| ||b|| sin x
can you explain this?
The cross product property is a mathematical rule that defines the relationship between two vectors in three-dimensional space. It states that the cross product of two vectors will result in a new vector that is perpendicular to both of the original vectors.
The cross product property is useful in many areas of science and engineering, such as physics, mechanics, and computer graphics. It allows us to calculate the direction and magnitude of a new vector based on the orientation of two other vectors.
The cross product is calculated using the formula: a x b = |a||b|sin(θ)n, where a and b are the two original vectors, |a| and |b| are their magnitudes, θ is the angle between them, and n is a unit vector perpendicular to both a and b.
No, the cross product property only applies to vectors in three-dimensional space. In higher dimensions, a similar concept called the wedge product is used instead.
The cross product property can be proven using geometric and algebraic methods. One proof involves using the dot product to show that the cross product is orthogonal to both original vectors. Another proof involves using the determinant of a matrix representation of the cross product to show its properties.