shape & space : vectors

vectors & scalars





Vectors & Scalars

A scalar is a quantity that has magnitude only.

e.g. mass, length, temperature, speed

A vector is a quantity with both magnitude and direction.

e.g. force, displacement, acceleration, velocity, momentum


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Vector notation


vector notation #3


The vector from X to Y may also be represented as V or -

vector notation

The magnitude of the vector(i.e. its number value) is expressed as:

vector notation #2


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Inverse vectors

An inverse vector is a vector of equal magnitude to the original but in the opposite direction.


vectors - inverse


vectors - inverse


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The Modulus(magnitude) of a vector

This modulus of a vector X is written l X l .

The modulus(length of the vector line) can be calculated using Pythagoras' Theorem.

This is dealt with in detail in the 'linear graphs section' here . However for completeness, the relevant formula is:


vectors - magnitude


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Scalar multiplication

A scalar quantity(i.e. a number) can alter the magnitude of a vector but not its direction.


vectors - scalar multiplication


Example - In the diagram(above) the vector of magnitude X is multiplied by 2 to become magnitude 2X.

If the vector X starts at the origin and ends at the point (4,4), then the vector 2X will end at (8,8).

The scalar multiplication can be represented by column vectors:


vectors - scalar multiplication


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The Triangle Law(Vector addition)

When adding vectors, remember they must run in the direction of the arrows(i.e head to tail).

A vector running against the arrowed direction is the resultant vector. That is, the one vector that would have the same effect as the others added together.


vectors triangle law #1



A and B are vectors, as shown below. Find the magnitude of their resultant X.

vectors example #1

First we must find the resultant vector. This is done by adding the column matrices representing the vectors.

vectors example #2


vectors example #1


vectors example #2


The magnitude of the resultant is given by using Pythagoras' Theorem:


vectors magnitude #2


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A single vector can be represented by two components set at 90 deg. to eachother. This arrangement is very useful in solving 'real world' problems.


vectors - components


looking at the right angled triangle below you can see where this came from




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Unit vectors

A unit vector has unit length (1).


vectors unit


the x-axis coordinate is i and the y-axis coordinate is j.


Example of a unit vector : 5i + 2 j would be at coordinates (5 , 2).


Unit vector addition (& subtraction):

In turn add i terms and then add j terms.


5 i + 2 j  plus  2 i + 5 j =  7 i + 7 j

in vector terms this can be expressed as:

vectors - unit addition


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