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

therefore,

**u v cos q = a x + b y + c z
q = cos-1 o (a x + b y + c z) / ( u v ) p**

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

**C _{x} = A_{x} + B_{x},**

**C _{y} = A_{y} + B_{y},**

**C _{z} = A_{z} + B_{z}**

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.

- Hold your right hand flat with your thumb perpendicular to your fingers. Don’t bend your thumb
- Point your fingers towards the direction of the first vector.
- Direct your palm so that when you fold your fingers they point in the direction of the second vector.
- Your thumb is pointing in the direction of the cross product.

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