12This gives the parametric equation of a line. are the direction numbers of the line
13Find the parametric equation of a line through the points (2, -1, 5) and (7, -2, 3)
14This gives the symmetric equation of a line. Solving for tThis gives the symmetric equation of a line.Write the line L through the point P = (2, 3, 5) and parallel to the vectorv=<4, -1, 6>, in the following forms:Vector functionParametricSymmetricFind two points on L distinct from P.
15We can obtain an especially useful form of a line if we notice that Substitute v into the equation for a line and reduce…
18Standard equation, general form, Functional form (*not in book) Equation of a PlaneStandard equation, general form, Functional form (*not in book)
19Scenario 1: normal vector and point Given any plane, there must be at least one nonzero vector n = <a, b, c> that is perpendicular to every vector v parallel to the plane.
20Standard Form or Point Normal Form By regrouping terms, you obtain the general form of the equation of a plane:ax+by+cz+d=0(Standard form and general form are NOT unique!!!)Solving for “z” will get you the functional form. (unique)
21Find the equation of the plane with normal n = <1, 2, 7> which contains the point (5, 3, 4). Write in standard, general, and functional form.
23Find the equation of the plane passing through (1, 2, 2), (4, 6, 1), and (0, 5 4) in standard and functional form.Note: using points in different order may result in a different normal and standard equation but the functional form will be the same.
24Does it matter which point we use to plug into our standard equation? Scenario 3: two linesDoes it matter which point we use to plug into our standard equation?
25Scenario 4: Line and a point not on line Find the equation of the plane containing the point (1, 2, 2) and the line L(t) = (4t+8, t+7, -3t-2)