Vectors #2: The Basics

Following on from the post on Monday, today I will be talking about the basic properties of vectors.

Vectors have a magnitude (size) and direction:

They are normally depicted using a bold alphabetical letter, e.g. a, or can be written as the letter of its initial and terminal position with an arrow above it.

The sum of any vectors a and b is:

a + b = (a1 + b1)i + (a2 + b2)j + (a3 + b3)k

This can be represented geometrically by placing the tail of the arrow b at the head of arrow b. The arrow that can be drawn from the tail of a to the head of is the resultant vector – ab.

This is sometimes referred to as the parallelogram rule as a and b form the sides of a parallelogram with aas one of its diagonals.

The subtraction of two vectors a and b is:

a – b = (a1 – b1)i + (a2 – b2)j + (a3 – b3)k

This can be represented geometrically as the line that connect the tail of b with the tail of a.

Length/Magnitude

This is denoted by ||a|| or |a| (do not confuse this with the absolute value!). The length of a vector can be calculated using Pythagoras’ theorem since the unit vectors i, j and k are orthogonal:

Unit Vector

A unit vector is any vector that has a magnitude of 1, and are therefore is used to indicate the direction. A unit vector, normally indicated with a hat â, can be calculated by ‘normalising’ a vector, that is, divide the vector by its magnitude:

Dot Product

The dot product of two vectors is denoted by a.b and is defined by:

where θ is the actue angle between a and b.

The dot product can also be calculated by adding the products of the components of each vector:

Geometrically, this can be described as drawing a and b with a common start point, and then multiplying the length of a with the length of that component of b that points in the same direction as a.

Cross Product

The cross product is only meaningful in 3 (or 7) dimensions and its result is a vector (unlike that of the dot product).

The cross product is denoted by a x b and is defined as being the vector perpendicular to both a and b. It can be calculated using the following equations:

where n is a unit vector perpendicular to both a and b;

or

The ‘right-hand’ rule is used to see which direction the cross product points, as it could point in the completely opposite direction:

Source: 1

I must admit that vectors are not by strongest area in mathematics, so it’s really interesting for me to do some research on them myself! Let me know what you think. M x