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:

Source: mathsisfun

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.

Addition and Subtraction

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.

The addition of two vectors a and b

Source: wikipedia

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.

The subtraction of two vectors a and b

Source: wikipedia

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:

Screen Shot 2016-06-03 at 6.12.11 PM.png

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:

Screen Shot 2016-06-03 at 6.14.48 PM

Dot Product

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

Source: maths.stackexchange

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:

Screen Shot 2016-06-03 at 6.19.02 PM

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).

2000px-Cross_product_parallelogram.svg.png

Source: wikipedia

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:

Screen Shot 2016-06-03 at 6.24.03 PM

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

or

Screen Shot 2016-06-03 at 6.25.50 PM

Source: omega.albany.edu

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

Source: mathsisfun

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

 

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