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## 2.7.2.2 Angle between two vectors:

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

### 2.7.2.3 Cross Product:

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.

2. Point your fingers towards the direction of the first vector.
3. Direct your palm so that when you fold your fingers they point in the direction of the second vector.
4. Your thumb is pointing in the direction of the cross product.

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