Basic procedure #
Define variables - What to solve for
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A system of equations - Write the equation that describe the problem
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Convert a matrix - Express your system of equations as an Augmented matrix M
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Solve - In matlab, insert the matrix as rref(M) and boom you almost have the answer - you do the rest clean work by your hands and leave the dirty work to matlab
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Interpret - Have the results, then interpret the answers in terms of the question
Curve fitting #
- Write the x-axis values in a row vector with a
'to convert to column vectors, then use.^to raise powers on each of their elements. - Then, horizontally contatenate them together with a column vector full of ones (
ones(dim, 1)) representing the constant term - Then add the y-axis value to the right and form a augmented matrix - then boom, you get what you want
Projection #
Easy calculation #
Calculate as the formula, here v projects onto u
$$ \text{proj}_uv = \frac{u\cdot v}{u\cdot u}u $$Perpendicular vector #
Line If were to calculate a perpendicular vector to a line, we calculate like this
$$ \text{t}=v-\text{proj}_uv $$Plane If were to calculate for a plane, we calculate two perpendicular vectors and add them up
Cartesian representation #
- We first calculate the normal vector using the approach above
- Then we substitute in a vector and get the
dinax + by + cz = d
Calculate intersection of planes #
- We first list the Cartesian forms together
- Then we do RREF on the system of equations, and see what we have left