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Lines and Planes in Space Day 2

Lines and Planes in Space Day 2. Shortest Distance between Skew Lines. Line 1 and Line 2 are skew. Suppose line 1 contains the point A and has direction vector u Suppose line 2 contains the point B and has direction vector v. is the shortest distance between the two lines.

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Lines and Planes in Space Day 2

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  1. Lines and Planes in Space Day 2

  2. Shortest Distance between Skew Lines Line 1 and Line 2 are skew Suppose line 1 contains the point A and has direction vector u Suppose line 2 contains the point B and has direction vector v is the shortest distance between the two lines.

  3. Ex. 1 Find the shortest distance between the skew lines: Line 1: A(0,1,2) Line 2: B(3,-1,4)

  4. The Vector Equation of a Plane To find the vector equation of a plane a point on the plane and two different direction vectors are required. The equation is defined as: where a is the fixed point on the plane and b and c are the vectors. In component form, this equation can be written as:

  5. An alternate way of defining a plane is to use the cross product of b and c. Let n = b x c (normal vector). n is perpendicular to any vector or line in the plane (or parallel to the plane) Let’s derive this formula: Suppose a plane contains fixed point A and has normal vector n. Let R be any point in the plane. is perpendicular to n. So:

  6. Let’s summarize the ways we can write the equation for a plane: 1) Since we can say which implies: 2) The Cartesian equation of the plane where is the normal vector, is a point on the plane, and d is a constant: (This is also called “normal form”)

  7. Ex. 2 Find the equation of the plane with normal vector and containing (-1, 2, 4). Use the point given and the normal vector to find the value for d:

  8. Ex. 3 Find the equation of the plane through A(-1, 2, 0), B(3, 1, 1) and C(1, 0, 3): a) in vector form and b) in normal form. a) For vector form, choose one point to be the fixed point, and find vectors between the others: Let’s choose C as the fixed point. We need two other non-parallel vectors: We could have chosen any point to be fixed and any combination of non-parallel vectors.

  9. A(-1, 2, 0), B(3, 1, 1), C(1, 0, 3) b) For normal form, again we can choose any of the three points to be the fixed point. Then find the normal using the vectors emanating from the fixed point to the two remaining points. Let’s choose A(-1,2,0) as the fixed point. Then:

  10. Ex. 4 Find the parametric equations of the line through A(-1,2,3) and B(2,0,-3) and hence find where (the point)this line meets the plane with equation: For the line, let’s use vector: and point A: (-1,2,3) Since the expressions for the parametric eqtn of the line represent any point on the line, substitute them into the equation for the plane: Solve for t: Substitute t into the line to find the point where the line and plane meet: (-7, 6, 15)

  11. Ex. 5 Find the coordinates of the foot of the normal from A(2,-1,3) to the plane . Hence find the shortest distance from A to the plane. Let’s call the point at the foot of the normal N. We need to find its coordinates. A(2,-1,3) Since we are given the normal form of the plane, the normal vector is: n N(x,y,z) Let’s write the parametric eqts for line AN using n as the direction vector: Sub these into the eqtn for the plane and solve for t:

  12. Ex. 6 Find the coordinates of the foot of the normal N from A(2,-1,3) to the plane with equation fixed pt vector direction vector direction vector Write the vector equation for the plane: A(2,-1,3) n N(x,y,z) Normal to the plane: (Normal vectors to the plane are normal to any point in the plane!) So eqtn for AN: Since N is a point on the line, it must have coordinates given by:

  13. Equation for the plane: Point in the plane: Solving with a calculator:

  14. Angles in Space Angle between a Line and a Plane n Line with direction d intersects a plane with normal n. The angle made with the line and normal is and the angle made with the line and the plane is . d Ex. 7 Find the acute angle between the plane and the line with equations:

  15. Angle between two planes The angle made between two planes is the same as the angle made by the planes respective normals. If and are the normals for the two planes, then the angle between the two planes is given by: (Notice this is simply the formula for the angle between two vectors- we’re just using the normals!)

  16. Ex. 8 Find the acute angle between the planes with equations: and

  17. The Intersection of Planes Two planes in space can be: 1) Intersecting 2) Parallel 3) Coincident

  18. Three planes in space can be: 1) Coincident 2)Two coincident and one other 3) Two coincident and one parallel 4) Two parallel and one other

  19. 5) All three parallel 6) All meet at one point 7) All three meet in a common line 8) The line of intersection of any 2 is parallel to the third plane.

  20. Ex. 9 Find the intersection of the planes: Since there are 3 variables, but only 2 eqtns, let z = t. (t is a real #) So: So… We can say that the solution to the system is given by: This is the line where the two planes meet! Let’s take this a step further…

  21. Substitute our solution to the first two into the third eqtn: Solve: This is the point where all three planes meet. (Unique solution)

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