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 Vector Properties Vector Algebra Vector Multiplication Differentiation

Vector Properties

It is the convention to print vectors in boldface type to distinguish them from scalars. Graphically speaking, an arrow can conveniently and effectively represent a vector. The length of the arrow is proportional to its magnitude while the tip of the arrow shows its direction (Figure 1).

Figure 1

A vector can also be specified by its components:

[1]

The number of the components basically determine the dimension of a vector: 2-dimensional (2 components) vs. 3-dimensional (3 components). The components are related to the coordinate system in use. See the Coordinate Systems page for the details. Figure 2 shows vectors described in Cartesian coordinates.

Figure 2

One can describe vector algebra in both graphically (using arrows) and with components.

The magnitude of the vector is the same to the length of the vector arrow shown in Figure 2:

[2]

A vector whose magnitude is unity (= 1) is called a unit vector. The unit vector of vector v shown in Figure 2 can be obtained by dividing the vector with its magnitude.

[3]

The unit vector of the axes (unit coordinate vectors) can be expressed as

[4]

where i, j & k = the unit vectors of the X, Y & Z axes, respectively.

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Vector Algebra

For two vectors a & b, let

[5]

If the two vectors are equal to each other, all the corresponding components are equal to each other as well:

[6]

Vectors of the same type can be added to yield the resultant vector. Adding two vectors is equivalent to adding the components (Figure 3):

Figure 3

[7]

As shown in Figure 3, vectors a and b were connected tip-to-tale, forming two sides of a triangle. The third side is the resultant vector (c) drawn from the tale of the first vector to the tip of the last vector added. This is the so-called tip-to-tale (graphical) approach of vector addition. The sequence of addition is not important in this process (commutative):

[8]

Addition of more than two vectors can be also performed similarly, two at a time:

[9]

As shown in [9], vector addition is associative. One can move the vectors freely to connect tip-to-tale as far as the magnitudes and the directions of the vectors are intact. The sequence in addition is not important.

When a vector is multiplied by a scalar, it magnifies (Figure 4):

Figure 4

[10]

where d = a scalar multiplier. The vector lengthens when |d| > 1 while it shortens if |d| < 1. If d is negative, the direction of the vector reverses. The operation shown in [10] is commutative:

[11]

Scalar multiplication is also distributive:

[12]

In order to compute the displacement or to compute change in velocity, one needs to subtract one vector from another of the same kind. Vector subtraction is equivalent to subtraction of the corresponding components (Figure 5):

[13]

Figure 5

Figure 5 shows two equivalent graphical methods that find c. As shown in [13], vector subtraction is a special case of vector addition.

Now, let's revisit Figure 2. As shown in Figure 6, a vector can be expressed as the sum of the component vectors:

Figure 6

[14]

where i, j & k = the unit coordinate vectors of the X, Y & Z axes, respectively.

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Vector Multiplication

There are two different kinds of vector multiplication: the scalar product and the vector product. First, the scalar product is defined as:

[15]

The result of the scalar product is a scalar. The scalar product is also called as the dot product following the product symbol used. The scalar product is commutative:

[16]

and distributive:

[17]

From [15]:

[18]

As shown in [18], square of the magnitude of a vector is the same to the scalar product of the vector with itself. Let's apply [18] to the triangle formed by the three vectors shown in Figure 7:

Figure 7

[19]

From the Pythagorean Theorem:

[20]

From [19] and [20]:

[21]

where q = the angle between the two vectors. [21] is an alternative definition of the scalar product. From [21]:

[22]

In other words, the components of a vector are its projections to the axes.

Another important and very useful vector multiplication is the vector product. The vector product is defined as:

[23]

It is understandable why it is called as the vector product: the result of this multiplication is a vector. It is also called as the cross product following the symbol used. From [23]:

[24]

and

[25]

From [23]:

[26]

In other words, the cross product of a vector with itself is always 0.

From [18], [21] and [23]:

[27]

One thing very interesting here is the fact that the resulting vector from the cross product is perpendicular to the two vectors involved in the cross product:

[28]

since

[29]

The actual direction of the resulting vector follows the right-hand rule: the direction of the resulting vector is the direction a right-handed screw advances when it is rotated from the direction of the first vector to the direction of the second vector through the smallest angle between them (Figure 8).

Figure 8

From the right-hand rule:

[30]

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Differentiation

Let vector v be a function of time (t):

[31]

The derivation operation is distributive so that the time-derivative of the vector is sum of the time-derivatives of the component vectors:

[32]

[32] holds because the unit coordinate vectors are constant. If the unit coordinate vectors are not constant (such as those in a rotating reference frame), [32] does not hold.

In general, the following rules also hold:

[33]

since

[34]

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 © Young-Hoo Kwon, 1998-