Dot Product Laws for Dot Product Scalar Projection Vector Projection Cross Product Laws for Cross Product Triple Product

Vector Multiplication

·                    Dot Product

Given two vectors "A" and "B", the dot product is defined as the product of the magnitude of A and the magnitude of B and the cosine of the smaller angle between them. The dot of scalar product of A and B is a scalar.

The following laws are valid for Dot Product:

1.      A · B = B · A

2.      A · (B + C) = A · B + A · C

3.      m(A · B) = (mA) · B

= A · (mB)

= (A · B)m

4.      i · i = j · j = k · k = 1

i · j = j · k = k · i = 0

5.      A · A  = A2

A · A = (Axax + Ayay + Azaz) · (Axax + Ayay + Azaz)

### |A|2 = {Ö(Ax 2 + Ay 2 + Az 2)}2 = Ax 2 + Ay 2 + Az 2

6.      If A = Ax i + Ay j + Az k and

B = Bx i + By j + Bz k

Then: A · B = Ax Bx + Ay By + Az Bz

7.      If A · B = 0, and A and B are not null vectors, then A and B are perpendicular.

#### ·                    Scalar Projection

-Scalar component of a vector in the direction of another vector

-The length of projection of a vector on another vector.

#### ·                    Vector Projection

-is the vector component of a vector in the direction of another vector.

·                    Cross Product

The cross product of two vectors say "A" and "B" is by definition:

where q is the smaller angle between "A" and "B" and an is a unit vector normal to the plane determined by "A" and "B" when they are drawn from a common point. The direction of an is the same as the direction of a right-hand screw where "A" is turned towards "B".

The following laws are valid for Cross Product:

1.      A x B = -B x A

2.      A x (B + C) = A x B + A x C

3.      m(A x B) = mA x B

= A x mB

= (A x B)m

4.      If A = Ax i + Ay j + Az k and

B = Bx i + By j + Bz k

Expansion by First Row:

Triple Products:

1.      (A · B)C ¹ A(B · C)

2.      A · (B x C) = B · (C x A)

=C · (A x B)

3.      A x (B x C) ¹ (A x B) x C

4.      A x (B x C) = (A · C) B - (A · B) C

(A x B) x C = (A · C) B - (B · C) A