3D-Planes (eActivity)

Command last updated 6/3/2024.

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Extends Charlie Watson’s eActivity. Modify the values of the input variables, then click the top line and hit EXE. The outputs are stored in the relevant variables.

Plane 3 pts

Find vector and Cartesian equations of plane given 1 point and 2 direction vectors on plane.

Input: A, B, C

A, B, C are point vectors on plane

Output: A, AB, AC soln, n, q, E

A is point vector on plane
AB, AC are direction vectors on plane
WARNING: If soln has 2 or more of x=x, y=y or z=z then ABC are collinear and infinite number of normals exist
n is normal vector
q is number
E is Cartesian equation

r = A + λAB + μAC

r·n = q (n = ai + bj + ck)

ax + by + cz = q

Example

Find a Cartesian equation of the plane containing points \((1, 0, 2)\), \((2, 3, 0)\) and \((1, 2, 1)\). (Sample Specialist Exam 2 Section B Question 5aii)

A = [1, 0, 2]; B = [2, 3, 0]; C = [1, 2, 1] A = [1, 0, 2]; AB = [1, 3, -2]; AC = [0, 2, -1]; soln = {x=z/2, y=z/2, z=z}; n = [1/2, 1/2, 1]; q = 5/2; E = x/2+y/2+z=5/2

Plane 1 pt, 2 vectors

Find Cartesian equation of plane given 1 point and 2 direction vectors on plane.

Input: A, P, Q

A is point vector on line
P, Q are direction vectors on plane

Output: soln, n, q, E

WARNING: If soln has 2 or more of x=x, y=y or z=z then P and Q are parallel vectors and infinite number of normals exist
n is normal vector
q is number
E is Cartesian equation

r = A + λP + μQ

r·n = q (n = ai + bj + ck)

ax + by + cz = q

Example

Find a Cartesian equation of the plane described by \(x = 1 + 2s + 3t\), \(y = -2 - s - 2t\) and \(z = 2 - s + t\). (Sample Specialist Exam 2 Section B Question 3b)

A = [1, -2, 2]; P = [2, -1, -1]; Q = [3, -2, 1] soln = {x=x, y=5x/3, z=x/3}; n = [1, 5/3, 1/3]; q = -5/3; E = x + 5y/3 + z/3 = -5/3

Plane Cartesian

Find vector equation of plane given Cartesian equation of plane.

Input: n, q

n is normal vector to plane
q is number

Output: A, P, Q

A is point vector on plane
P, Q are direction vectors on plane

ax + by + cz = q

r·n = q (n = ai + bj + ck)

r = A + λP + μQ

Example

Find a vector equation of the plane \(2x + 3y - 5z = 7\).

n = [2, 3, -5]; q = 7 A = [7/2, 0, 0]; P = [5/2, 0, 1]; Q = [-3/2, 1, 0]

Intersection plane-line

Find intersection between line and plane.

Input: a, b, n, p

a is point vector on line
b is direction vector along line
n is normal vector to plane
p is point vector on plane

Output: X

X is point vector of intersection

r = a + tb

r·n = p·n

Example

Find the point of intersection between the plane \(2x - 2y - z = -18\), and a line normal to the plane passing through the origin. (2023 Specialist Exam 2 Section B Question 5f)

a = [0, 0, 0]; b = [2, -2, -1]; n = [2, -2, -1]; p = [0, 0, 18] X = [-4, 4, 2]

Intersection plane-plane

Find intersection between two planes.

Input: P1, n1, P2, n2

P1 is point vector on plane 1
n1 is normal vector to plane 1
P2 is point vector on plane 2
n2 is normal vector to plane 2

Output: A, B

A is point vector on intersection line
B is direction vector along intersection line

r·n1 = P1·n1

r·n2 = P2·n2

r = A + λB

Example

Find the line of intersection between the planes \(x + y + 2z = 5\) and \(x - y - z = 0\). (Sample Specialist Exam 2 Section B Question 5bii)

P1 = [5, 0, 0]; n1 = [1, 1, 2]; P2 = [0, 0, 0]; n2 = [1, -1, -1] A = [0, -5, 5]; B = [1, 3, -2]

Distance plane-point

Find shortest distance between point and plane.

Input: A, P, n

A is point vector of point
P is point vector on plane
n is normal vector to plane

Output: D

D is shortest distance

r·n = P·n

Example

Find the shortest distance between the origin and the plane \(2x - 2y - z = -18\). (2023 Specialist Exam 2 Section B Question 5e)

A = [0, 0, 0]; P = [0, 0, 18]; n = [2, -2, -1] D = 6

Distance plane-line

Find shortest distance between line and plane.

Input: A, B, P, n

A is point vector on line
B is direction vector along line
P is point vector on plane
n is normal vector to plane

Output: D

D is shortest distance

r = A + λB

r·n = P·n

Example

Find the shortest distance between the line \(\mathbf{r}(t) = \mathbf{i} + \mathbf{j} - 5\mathbf{k} + t(4\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\) and the plane \(2x - 3y - z = 2\). (Sample Specialist Exam 2 Section B Question 4c)

A = [1, 1, -5]; B = [4, 2, 2]; P = [1, 0, 0]; n = [2, -3, -1] D = sqrt(14)/7

Distance plane-plane

Find shortest distance between two planes.

Input: P1, n1, P2, n2

P1 is point vector on plane 1
n1 is normal vector to plane 1
P2 is point vector on plane 2
n2 is normal vector to plane 2

Output: D

D is shortest distance

r·n1 = P1·n1

r·n2 = P2·n2

Example

Find the shortest distance between the planes \(5x - 4y - 12z = 10\) and \(-15x + 12y + 36z = 20\). (Sample Specialist Exam 2 Section A Question 6)

P1 = [2, 0, 0]; n1 = [5, -4, -12]; P2 = [0, 5/3, 0]; n2 = [-15, 12, 36] D = 10 sqrt(185) / 111

Angle plane-line

Find both angles between line and plane.

Input: b, n

b is direction vector along line
n is normal vector to plane

Output: θ1, θ2

θ1, θ2 are angles

r = a + λb

r·n = k

Example

Find the acute angle at which the line \(\mathbf{r}(t) = 3\mathbf{i} + 2\mathbf{j} + 4\mathbf{k} + t(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})\) intersects the plane \(2x - 2y - z = -18\). (2023 Specialist Exam 2 Section B Question 5c)

b = [1, -2, 2]; n = [2, -2, -1] theta1 = 26; theta2 = 154

Angle plane-plane

Find both angles between two planes.

Input: n1, n2

n1 is normal vector to plane 1
n2 is normal vector to plane 2

Output: θ1, θ2

θ1, θ2 are angles

r·n1 = k1

r·n2 = k2

Example

Find the value of \(k\) for which the planes \(2x - ky + 3z = 1\) and \(2kx + 3y - 2z = 4\) are perpendicular. (2023 Specialist Exam 2 Section A Question 18)

n1 = [2, -6, 3]; n2 = [2*6, 3, -2] theta1 = 90; theta2 = 90

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