Let the coordinates of any two nonzero vectors u and v. The angle q between them:
u = ai + bj + ck
v = xi + yj + zk
u.v = u v cos q
u.v = a x + b y + c z
The vector product of two vectors A and B, also called the cross product, is denoted by A X B.
The symbol used to represent cross product is a cross (×). Since this product has both, magnitude and direction, it is also known as the vector product.
A × B = AB sin θ ˆn
The vector ˆn ("n hat") is a unit vector perpendicular to the plane formed by the two vectors. The direction of ˆn is determined by the right hand rule.
The cross product is distributive i.e.
A × (B + C) = (A × B) + (A × C)
but not commutative i.e.
A × B = −B × A
On reversing the order of cross multiplication the direction of the product also get reversed
If we know the components of A and B, we can calculate the components of the vector product, using a procedure similar to that for the scalar product.
The components of
C = A X B
are given by
Cx = Ax + Bx,
Cy = Ay + By,
Cz = Az + Bz
The cross product of any vector with itself is zero .i.e.
A × A = 0
Applying this corollary to the unit vectors means that the cross product of any unit vector with itself is zero.
î × î = ĵ × ĵ = ˆk × ˆk = (1)(1)(sin 0°) = 0
It should be noted that the cross product of any unit vector with other will give a magnitude of one, because
sine90° = 1
The right hand rule for cross multiplication relates the direction of the two vectors with the direction of their product. Since cross multiplication is not commutative.
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