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When dealing with vectors,it is possible to define multiplaication in a number of ways. For example, in performing scalar multiplications,we simply multiplay each component of a vector by the scalar.For example,for a constant A and a vector(x,y,z):


But what are we to do when we are asked to multiply two vectors together? In this class you will make use of two common definitions:the dot product and the cross product.Each definition is usefull because of its properties.

The Dot Product

In Catesian coordinates, the dot product of two vectors is the sum of the component-wise products.
For example:


The dot product can also be defined in an equivalent manner geometrically. In this case the dot product of two vectors is equal to the product of the magnitude of each vector times the cosine of the smallest angle between them. For two vectors u and v:

theta=smallest angle between u and v

Properties of the Dot Product:
The dot product always returns a scalar number,not a vector
The dimension of the vectors involved in the dot product must match
Using the geometric definition, we can see that the dot product of two vectors is equal to the component of the first vector that is in the direction of the second multiplied by the magnitude of the second vector
Both descriptions together allow us to compute the angle theta given the value of two vectors u and v:
                                   /          u * v        
            theta=arc cos | ---------------- |
                                   ||u||||v||    /
Since the cosine 90degree equals 0, two perpendicular vectors will always have a dot product of 0.

The Cross Product

In cartesian coodinates, the cross product can be described in two ways. It can be described as the determinant of matrix or its magnitude can be described geometrically. In determinant form, two vectors v=(a,b,c) and w=(d,e,f) yield:
                   _ _ _
                 |x y z|     /                                   
v W = det |a b c| = | (bf-ec),-(af-dc),(ae-db) |
                 |d e f|                                         /

Alternatively, if we are only interested in the magnitude of the cross product :
||v w|| =||v|| ||w|| sin(theta)
theta=smallest angle between the vectors

Properties of the Cross Product:
The cross product returns a vector quantity, not a scalar
Two vectors in the same or opposite direction will have a cross product equal to zero, since the sine of both 0 and 180 degrees is zero
The magnitude of the cross product of vectors v and w may be thought of as geometrically representing the area of the parallelogram that can be defined from the vectors.
The direction of the resulting vector will always be perpendicular to both of the original vectors, and it will be in the direction given by the right-hand-rule.

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