1837.sagews

a=vector([1,0,5]);a  #initialize vector in R^3

(1, 0, 5)

b=vector([-2,1,0]);b  #initialize vector in R^3

(-2, 1, 0)

c=a.cross_product(b);c  #find vector cross product

(-5, -10, 1)

area1=c.norm();area1  #magnitude of axb is area of parallelogram described by a and b

3*sqrt(14)

area1.n(digits=10)  #magnitude of axb is area of parallelogram described by a and b

11.22497216

area2=abs(c);area2  #magnitude of axb is area of parallelogram described by a and b

3*sqrt(14)

area2.n(digits=100)  #magnitude of axb is area of parallelogram described by a and b

11.22497216032182415675124619694964790526805942333618083891123640196010546892081708393042968558313871

theta1=asin(area1/norm(a)/norm(b));theta1.n()  #find angle between a and b a la torque

1.39447248797915

theta2=theta1*180/pi;theta2.n()  #find angle between a and b a la torque

79.8973882083124

(a.plot(rgbcolor='red')+b.plot(rgbcolor='blue')+c.plot(rgbcolor='green')).show(aspect_ratio=1)  #c is orthogonal to a and b

d=a.dot_product(b);d  #scalar product

-2

theta3=acos(d/norm(a)/norm(b));theta3.n()  #find angle between a and b in the parallelogram

1.74712016561065

theta4=theta3*180/pi;theta4.n()  #find angle between a and b in the parallelogram

100.102611791688

e=vector([1,1,1]);e  #initialize another vector in R^3

(1, 1, 1)

vol=abs(e.dot_product(c));vol  #find volume of parallelepiped described by a, b and e

14

f=2*a;f  #dilate a (multiply by scalar)

(2, 0, 10)

g=-3*b;g  #dilated b (multiply by scalar)

(6, -3, 0)