.TL
Vectors in comp sci
.AU
Lucas Standen
.AI
QMC
.2C

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delim @@
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delim @#
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.NH 1
How to write them 

.LP
To write  a vector, like in maths we can use 
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({i sub x, j sub y})
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But they can also be written
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R sup 2
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R sup 3
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Where the power is the number of degrees available

.NH 1
Combining vectors
.LP
To combine vectors one can use the formula
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w = alpha u + beta v
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Where w is the combined vector and
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alpha + beta = 1
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.NH 2
Example
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u = (2,2)
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v = (6,-2)
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We can then say that
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w = (4, 0)
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By subtracting v from u

Then using the formula
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2 alpha + 6 beta = 3
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Where 3 is a point on the combined vector
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2 alpha + -2 beta = 1
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We can then solve for @ beta # like so

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6 beta - 3  = -2 beta - 1
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8 beta - 2 = 0
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8 beta = 2
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beta = 2 over 8
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beta = 1 over 4
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From this we can say 
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alpha = 3 over 4
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Because 
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alpha + beta = 1
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.NH 2
Another example

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2 alpha + 6 beta = 2
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2 alpha - 2 beta = 1
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8 beta = 1
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beta = 1 over 8
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2 alpha - 2 ({1 over 8}) = 1
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2 alpha = 5 over 4
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alpha = 5 over 8
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Since 
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alpha + beta != 1
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We can say that w does not lie on the vector uv

And because it is greater than 1 it means it is inside the triangle created by u and v

.NH 1
The dot product

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To solve use the following formula

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u.v = |u|.|v| cos( theta )
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Where @ theta # is the angle between the 2 vectors and

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|u| = " magnitude of u, " sqrt {x sup 2 + y sup 2}
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You can also use
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u.v = u sub 1 . v sub 1 + u sub 2 . v sub 2 + u sub n + v sub n ...
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If you don't have the angle

Don't be confused by the dot, it just means
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u sub 1 . v sub 1 = u sub 1 times v sub 1
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.NH 2
Exam question

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1.1)
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|b| = 4
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1.2)
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u.v = u sub 1 . v sub 1 + u sub 2 . v sub 2 + u sub n + v sub n ...
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a.b =  4 . 4 + 3 . 0
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a.b =  16 + 0
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a.b =  16 
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