a = vector([1, 3, 4])
b = vector([2, 7, -5])
print b.cross_product(a)
(43, -13, -1) (43, -13, -1) |
a = vector([1, 2, -2])
b = vector([3, 0, 1])
print a.inner_product(a.cross_product(b))
print b.inner_product(a.cross_product(b))
0 0 0 0 |
P = vector([1, 4, 6])
Q = vector([-2, 5, -1])
R = vector([1, -1, 1])
a = Q - P
b = R - P
u = a.cross_product(b)
print u / u.norm()
(-4/41*sqrt(82), -3/82*sqrt(82), 3/82*sqrt(82)) (-4/41*sqrt(82), -3/82*sqrt(82), 3/82*sqrt(82)) |
P = vector([1, 4, 6])
Q = vector([-2, 5, -1])
R = vector([1, -1, 1])
a = Q - P
b = R - P
show((1/2)*(a.cross_product(b)).norm())
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a = vector([1, 4, -7])
b = vector([2, -1, 4])
c = vector([0, -9, 18])
print a.inner_product(b.cross_product(c))
0 0 |
var('a, b, c')
v = vector([a, b, c])
p = vector([1, 2, 1])
q = vector([3, 1, -5])
pv = p.cross_product(v)
solve([pv[0] == q[0], pv[1] == q[1], pv[2] == q[2]], a, b, c)
[[a == r2 + 1, b == 2*r2 - 3, c == r2]] [[a == r2 + 1, b == 2*r2 - 3, c == r2]] |
w = vector([1, -3, 0])
p.cross_product(w) == q
True True |
range(0,3)
[0, 1, 2] [0, 1, 2] |
t = var('t')
r0 = vector([5, 1, 3])
v = vector([1, 4, -2])
r = r0 + t*v
print r
(t + 5, 4*t + 1, -2*t + 3) (t + 5, 4*t + 1, -2*t + 3) |
t = var('t')
A = vector([2, 4, -3])
B = vector([3, -1, 1])
r0 = A
v = B-A
r = r0 + t*v
print r
(t + 2, -5*t + 4, 4*t - 3) (t + 2, -5*t + 4, 4*t - 3) |
var('t, s')
L1 = parametric_plot3d((1 + t, -2+3*t, 4 - t), (t, -50, 50), color="green")
L2 = parametric_plot3d((2*s, 3 + s, -3 + 4*s), (s, -50, 50), color="red")
L1 + L2
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Sleeping...
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var('t, s')
solve([1 + t == 2*s, -2 + 3*t == 3+s], t, s)
[[t == (11/5), s == (8/5)]] [[t == (11/5), s == (8/5)]] |
t = 11/5
s = 8/5
bool(4 - t == -3 + 4*s)
False False |
u = vector([3, 2, -1])
Q = vector([2, 1, -1])
P = vector([0, 1, -5])
PQ= Q-P
(PQ.cross_product(u)).norm() / u.norm()
1/7*sqrt(69)*sqrt(14) 1/7*sqrt(69)*sqrt(14) |
var('x, y, z')
p = vector([2, 4, -1])
n = vector([2, 3, 4])
q = vector([x, y, z])
n.inner_product (q - p) == 0
2*x + 3*y + 4*z - 12 == 0 2*x + 3*y + 4*z - 12 == 0 |
implicit_plot3d(2*x + 3*y + 4*z - 12 == 0, (x, -3, 3), (y, -3, 3), (z, -3, 3))
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Sleeping...
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solve(2*x + 3*y + 4*z - 12 == 0, z)
[z == -1/2*x - 3/4*y + 3] [z == -1/2*x - 3/4*y + 3] |
var('x, y, z')
P = vector([1, 3, 2])
Q = vector([3, -1, 6])
R = vector([5, 2, 0])
T = vector([x, y, z])
n = (Q - P).cross_product(R - Q)
n.inner_product (T - P) == 0
12*x + 20*y + 14*z - 100 == 0 12*x + 20*y + 14*z - 100 == 0 |
var('x, y, z')
n1 = vector([1, 1, 1])
n2 = vector([1, -2, 3])
ang_rad = arccos((n1.inner_product(n2))/(n1.norm()*n2.norm())).n(digits=5)
print ang_rad
1.2571 1.2571 |
n1.inner_product(n2)
2 2 |
solve([x + y + z == 1, x - 2*y + 3*z == 1], x, y, z)
[[x == -5/3*r3 + 1, y == 2/3*r3, z == r3]] [[x == -5/3*r3 + 1, y == 2/3*r3, z == r3]] |
n1 = vector([1, 1, 1])
n2 = vector([1, -2, 3])
n1.cross_product(n2)
(5, -2, -3) (5, -2, -3) |
P = vector([1/2, 0, 0])
n = vector([5, 1, -1])
d = -1
D = abs(n.inner_product(P) + d) / n.norm()
print D
1/6*sqrt(3) 1/6*sqrt(3) |
u1 = vector([1, 3, -1])
u2 = vector([2, 1, 4])
P = vector([1, -2, 4])
Q = vector([0, 3, -3])
n = u1.cross_product(u2)
T = vector([x, y, z])
n.inner_product (T - Q) == 0
13*x - 6*y - 5*z + 3 == 0 13*x - 6*y - 5*z + 3 == 0 |
d = 3
D = abs(n.inner_product(P) + d) / n.norm()
print D
4/115*sqrt(230) 4/115*sqrt(230) |
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