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Coursework Notes - Shape & Space




vectors & scalars

vector notation

triangle law


unit vectors

inverse vectors

scalar multiplication




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






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 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 vector in terms of unit vectors : 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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