Vector Differentiation

In calculus we compute derivatives of real functions of a real variable. In the case of functions of a single variable

            y = f(x)

we compute the derivative of y with respect to x. In the case of a function of several variables

            y = f(x₁, x₂, ... , x_n)

we can compute the derivative of y with respect to any one of the independent variables, viewing the others as fixed.

In vector analysis we compute derivatives of vector functions of a real variable; that is we compute derivatives of functions of the type

            F(t) = f₁(t) i + f₂(t) j + f₃(t) k

This function can be viewed as describing a space curve. Intuitively it can be regarded as a position vector, expressed as a function of t, that traces out a space curve with increasing values of t. Expressed in different notation it is the function
      R(t) = x(t) i + y(t) j + z(t) k

Vector Multiplication: Cross Product

We define the cross product of two three-dimensional vectors a and b by the requirements:

• a × b is a vector that is perpendicular to both a and b.
• ||a × b|| is the area of the parallelogram spanned by a and b (i.e. the parallelogram whose adjacent sides are the vectors a and b).
• The direction of a×b is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from a to b, then the thumb points in the direction of a × b.)

Vector Multiplication: Dot Product

The multiplication of two vectors, is not uniquely defined, in the sense that there is a question as to whether the product will be a vector or not. For this reason there are two types of vector multiplication.
First, the scalar or dot product of two vectors, which results in a scalar.
And secondly, the vector or cross product of two vectors, which results in a vector.

In this tutorial we shall discuss only the scalar or dot product.