12 This gives the parametric equation of a line. are the direction numbers of the line
13 Find the parametric equation of a line through the points (2, -1, 5) and (7, -2, 3)
14 This 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.
15 We can obtain an especially useful form of a line if we notice that Substitute v into the equation for a line and reduce…
18 Standard equation, general form, Functional form (*not in book) Equation of a PlaneStandard equation, general form, Functional form (*not in book)
19 Scenario 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.
20 Standard 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)
21 Find 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.
23 Find 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.
24 Does 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?
25 Scenario 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)