Lines and Planes
Tools≻Load Package: Student Multivariate Calculus
Example 1: Equation of a Plane
Obtain the equation of the plane containing the three points 1,2,3, −1,3,1, 2,1,−1.
Write a sequence of the three points.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
In the "Choose Variables for Plane: dialog, accept default names or provide new ones.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
1,2,3,−1,3,1,2,1,−1→make plane<< Plane 1 >>→representation−6⁢x−10⁢y+z=−23
Example 2: Skew Lines
Show that x=1+2 t,y=2−3 t,z=3+5 t and x=3−s,y=5+3,z=7+6 s define skew lines, and find the distance between them.
Create Line Objects for each line
Form a list of the parametric equations defining a line.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line≻t or s, as appropriate
Context Panel: Assign to a Name≻L1 (or L2, as appropriate)
x=1+2 t,y=2−3 t,z=3+5 t→make line<< Line 1 >>→assign to a nameL1
x=3−s,y=5+3,z=7+6 s→make line<< Line 2 >>→assign to a nameL2
Verify the lines are skew
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Skew (or Parallel or Intersects)
Obtain the distance between the lines
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
Context Panel: Approximate≻10 (digits)
L1,L2→distance75311⁢622→at 10 digits6.014452050
The standard approach to finding the distance between skew lines is vectorial: Obtain N, the vector orthogonal to both lines, and project V, any vector from one line to the other, onto N. The length of this projection is the distance between the lines.
Obtain N, the common normal
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Direction
Context Panel: Assign to a Name≻V1 (or V2, as applicable)
L1→direction2−35→assign to a nameV1
L2→direction−106→assign to a nameV2
Common-Symbols palette: Cross-product operator
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻N
V1×V2 = −18−17−3→assign to a nameN
Obtain V, a vector from one line to the other