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.TL
Vectors in comp sci
.AU
Lucas Standen
.AI
QMC
.2C
.EQ
delim @@
.EN
.EQ
delim @#
.EN
.NH 1
How to write them
.LP
To write a vector, like in maths we can use
.EQ
({i sub x, j sub y})
.EN
But they can also be written
.EQ
R sup 2
.EN
.EQ
R sup 3
.EN
Where the power is the number of degrees available
.NH 1
Combining vectors
.LP
To combine vectors one can use the formula
.EQ
w = alpha u + beta v
.EN
Where w is the combined vector and
.EQ
alpha + beta = 1
.EN
.NH 2
Example
.EQ
u = (2,2)
.EN
.EQ
v = (6,-2)
.EN
We can then say that
.EQ
w = (4, 0)
.EN
By subtracting v from u
Then using the formula
.EQ
2 alpha + 6 beta = 3
.EN
Where 3 is a point on the combined vector
.EQ
2 alpha + -2 beta = 1
.EN
We can then solve for @ beta # like so
.EQ
6 beta - 3 = -2 beta - 1
.EN
.EQ
8 beta - 2 = 0
.EN
.EQ
8 beta = 2
.EN
.EQ
beta = 2 over 8
.EN
.EQ
beta = 1 over 4
.EN
From this we can say
.EQ
alpha = 3 over 4
.EN
Because
.EQ
alpha + beta = 1
.EN
.NH 2
Another example
.EQ
2 alpha + 6 beta = 2
.EN
.EQ
2 alpha - 2 beta = 1
.EN
.EQ
8 beta = 1
.EN
.EQ
beta = 1 over 8
.EN
.EQ
2 alpha - 2 ({1 over 8}) = 1
.EN
.EQ
2 alpha = 5 over 4
.EN
.EQ
alpha = 5 over 8
.EN
Since
.EQ
alpha + beta != 1
.EN
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
.LP
To solve use the following formula
.EQ
u.v = |u|.|v| cos( theta )
.EN
Where @ theta # is the angle between the 2 vectors and
.EQ
|u| = " magnitude of u, " sqrt {x sup 2 + y sup 2}
.EN
You can also use
.EQ
u.v = u sub 1 . v sub 1 + u sub 2 . v sub 2 + u sub n + v sub n ...
.EN
If you don't have the angle
Don't be confused by the dot, it just means
.EQ
u sub 1 . v sub 1 = u sub 1 times v sub 1
.EN
.NH 2
Exam question
.LP
1.1)
.EQ
|b| = 4
.EN
1.2)
.EQ
u.v = u sub 1 . v sub 1 + u sub 2 . v sub 2 + u sub n + v sub n ...
.EN
.EQ
a.b = 4 . 4 + 3 . 0
.EN
.EQ
a.b = 16 + 0
.EN
.EQ
a.b = 16
.EN
|
