Notes Lines 2

12.6 Lines Vector form: r(t) = (x0 i + y0 j + z0 k) + t(d1i + d2 j + d3k) x(t) = x0 + td1 Scalar form: y(t) = y0 + td2 z(t) = z0 + td3 Symmetric form: x − x0 y − y0 z − z0 = = d1 d2 d3 Examples: 1) Find a vector parameterization for the line that passes through P(3, 2, 3) and is parallel to the line r(t) = (i + j - k) + t(2i + 3j - 3k). 2) Give the answer to #1 in scalar form 3) Give the answer to #1 in symmetric form 4) Find a set of scalar parametric equations for the line that passes through P(2, 2, 1) and Q(3, -2, -2). 5) Find a set of scalar parametric equations for the line that passes through P(1, 2, -3) and is perpendicular to the xz-plane. Popper 1 6. A direction vector for the line a. b. c. d. e. x−4 y+3 = = z − 1 is −2 5 d = (4, -3, 1) d = (-2, 5, 0) d = (2, -5, -1) d = (-2, 5, 1) None of the above Parallel lines have direction vectors that are scalar multiples of each other. 6) Give a vector parameterization for the line that passes through P(1, 2, -2) and is parallel to the line: 3(x − 1) = 2(y − 2) = 6(z + 2) 7) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection: Two lines (l1: r(t) = r0 + t d and l2: R(u) = R0 + u D) intersect if there exists numbers t and u such that r(t) = R(u) 8) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection: If two lines are not parallel and do not intersect, then they are skew. 9) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection: Popper 1 7. The lines 1 : x = −2 + 4t, y = 3 − 2t, z = 3 + 6t and  2 : x = 6 − 2s, y = −1+ s, z = 3 − 3s are a. b. c. d. e. Coincident Parallel but not coincident Have a unique point of intersection Skew None of the above To find the angle between two lines: cosθ = ud • uD Distance from a point to a line: d (P, l ) = P0 P1 × d d 10) Find the distance from P(2, 0, 2) to the line through P0(3, -1, 1) parallel to i - 2j - 2k. Popper 1 8. The cosine of the angle between the lines 1 :  2 : x = 2 − 3s, y = −4 + s, z = 1+ 4s is a. b. c. d. e. −19 26 17 1 26 17 19 26 17 15 26 17 None of these x + 2 y − 2 z +1 and = = 3 −2 −2 12.7 Planes Let N = Ai + Bj + Ck be the nonzero vector perpendicular to a plane: The equation of the plane containing P(x0 , y0 , z0 ) with normal N = Ai + Bj + Ck is A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0 Examples: 1) Find an equation for the plane which passes through the point P(2, 4, 0) and is perpendicular to i - 3j + 2k. 2) Find an equation for the plane which passes through the point P(3, -2, 3) and is parallel to the plane: 3) Find an equation for the plane which passes through the point P(1, 3, 4) and contains the line: 4) Find the unit normals to the plane: 5) Write the equation of the plane in intercept form and find the points where it intersects the coordinate axes. Popper 1 9. An equation for the plane which passes through the point P(-3, 2, -4) and is perpendicular to x+2 z−5 the line is = y−3= 6 −2 a. 6(x + 3) − 2(z + 4) = 0 b. 6x + y − 2z = 0 c. 6(x − 3) + (y + 2) − 2(z − 4) = 0 d. 6(x + 3) + (y − 2) − 2(z + 4) = 0 e. None of these Angle between two planes: cos θ = u N 2 • u N 2 6) Find the angle between the planes. and Distance from a point to a plane d(P1 ,℘) = | Ax1 + By1 + Cz1 + D | A2 + B2 + C 2 7)Find the distance from the point P(1, -1, 2) to the plane 8) Find an equation in x, y, z for the plane that passes through the given points. P(1, -1, 2), Q(3, -1, 3), R(2, 0, 2) Two planes will be parallel if their norms are scalar multiples of each other. If two planes are not parallel, then they will intersect in a line. The line of intersection will have a direction vector equal to the cross product of their norms. 9) Find a set of scalar parametric equations for the line formed by the two intersecting planes. p1 : x + 2y + 3z = 0, p2 : 3x − 4y − z = 0 . Popper 1 10. A direction vector for the line of intersection of the planes x − y + 2z = −4 and 2x + 3y − 4z = 6 is a. d = i − j + 5k b. d = −2i + 8 j + 5k c. d = −2i + 8 j + k d. d = −10i − 8 j − 5k e. None of these