py import symbols, Matrix, acos, sqrt, pi, simplify

Vectors of the lines

A = Matrix([1, 2, 2]) # Direction vector of the first line B = Matrix([3, 2, 6]) # Direction vector of the second line

Calculate the angle between the lines

dot_product = A.dot(B) magnitude_A = sqrt(A.dot(A)) magnitude_B = sqrt(B.dot(B)) cos_theta = dot_product / (magnitude_A * magnitude_B) theta = acos(cos_theta)

Cross product to find vector perpendicular to both lines

cross_product = A.cross(B)

Unit vector perpendicular to both lines

unit_vector = cross_product / sqrt(cross_product.dot(cross_product))

Form the equation of the line perpendicular to both lines and passing through (1, 1, 1)

The parametric equation of the line is r = (1, 1, 1) + t * (unit_vector)

t = symbols('t') point = Matrix([1, 1, 1]) line_equation = point + t * unit_vector

theta_deg = theta * 180 / pi # Converting the angle to degrees

theta, theta_deg, unit_vector,### Answers: a) Angle between the lines: $\cos^{-1}\left(\frac{19}{21}\right)$ or approximately $\frac{180}{\pi} \cos^{-1}\left(\frac{19}{21}\right)$ degrees.

b) Unit vector perpendicular to both lines: $\frac{2\sqrt{5}}{5}\mathbf{i} + 0\mathbf{j} - \frac{\sqrt{5}}{5}\mathbf{k}$.

c) Equation of the perpendicular line passing through (1,1,1):
$\mathbf{r} = (1 + \frac{2\sqrt{5}t}{5})\mathbf{i} + 1\mathbf{j} + (1 - \frac{\sqrt{5}t}{5})\mathbf{k}.$

Let me know if you need detailed steps or further clarifications!
Here are some related questions:

1. How can we compute the angle if the cosine value is known?
2. How does the cross product help find a vector perpendicular to two lines?
3. What is the significance of unit vectors in geometry?
4. How do parametric equations represent straight lines in 3D?
5. Can you use the scalar triple product to verify coplanarity of vectors?

Tip: Always normalize a vector before using it