CalculusMadeEasy_CH1

1-6

a=np.array([[1],[2],[3]])
la.norm(a)
3.7416573867739413

b=np.array([[-1],[-3],[-4]])
la.norm(b)
5.0990195135927845

c=np.array([[1],[-2],[4]])
la.norm(c)
4.58257569495584

1-7

a=np.array([[1],[2],[-3]])
b=np.array([[-4],[-3],[6]])
ab_dot=np.dot(a.T, b)
ab_dot
array([[-28]])
cos_ab=ab_dot/(la.norm(a)*la.norm(b))
cos_ab
array([[-0.95814028]])
rad=np.arccos(cos_ab)
rad
array([[2.85123057]])
theta=np.rad2deg(rad)
theta
array([[163.36347829]])

1-8

t=symbols("t")
A=Matrix([[2,2],[-3,-3],[4,4]])
B=Matrix([[2,1],[4,-1],[5,-1]])
x=Matrix([[1],[t]])
A*x
$\left[\begin{matrix}2 t + 2\\- 3 t - 3\\4 t + 4\end{matrix}\right]$
B*x
$\left[\begin{matrix}t + 2\\4 - t\\5 - t\end{matrix}\right]$
eq=(A*x).T*(B*x)
eq
[(4−𝑡)(−3𝑡−3)+(5−𝑡)(4𝑡+4)+(𝑡+2)(2𝑡+2)]
eq=expand(eq)
eq
[𝑡2+13𝑡+12]
solve(eq)
[{t: -12}, {t: -1}]