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Transcript

MTH 212 Calculus III Notes 11.5 Lines and Planes in Space Consider a line in space through the point P( x1 , y1 , z1 ), in the direction of v = <a, b, c> (the direction vector). If the point Q (x, y, z) is on this line, then the vector PQ must be parallel with the direction vector v. Thus we can write: PQ = t v for some scalar t. This means: PQ = x x1 , y y1 , z z1 at , bt , ct = tv Since these two vectors are equal, their corresponding components are equal, and we obtain the parametric equations of a line in space: x x1 at, y y1 bt, z z1 ct for t a real number. If you solve each of these parametric equations for t, and the direction numbers a, b, and c are nonzero, we can write the symmetric equations of a line in space: Ex. Find a set of parametric equations and the corresponding set of symmetric equations of the line that passes through the points (–5, 2, 3) & (3, 1, 0). Planes in Space The equation of a plane can be obtained from a point on the plane and a vector normal (perpendicular) to it. Consider a plane containing the point P( x1 , y1 , z1 ), with the normal vector n = <a, b, c>. Then the plane consists of all points Q(x, y, z) such that the vector PQ is orthogonal to n. This means that for any point Q(x, y, z) on the plane, n · PQ = 0 Ex. Find the equation of the plane containing the points (2, 3, -2), (3, 4, 2) & (1, -1, 0).