- parametric equations for the line, z z 0 td 3 or if d 1 0, d 2 0andd 3 0, - symmetric equations of the line.

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1 Lines and Planes in Space -(105) Questions: What do we need to know to determine a line in space? What are the fms of a line? If two lines are not parallel in space, must they be intersect as two lines in plane? What do we need to know to determine a plane in space? 1 Lines in Space Consider a line which passes through the point P 0 x 0, y 0, z 0 in the direction d d 1, d 2, d LetP x,y,z be a point on the line Then the vect P 0 P is parallel to the direction vect d,thatis,p 0 P t d where t is a scalar Hence, P 0 P x x 0, y y 0, z z 0 t d td 1, td 2, td x x 0 td 1 y y 0 td 2 - parametric equations f the line, z z 0 td if d 1 0, d 2 0andd 0, x x 0 y y 0 - symmetric equations of the line d 1 d 2 d In the case where d 1 0, the line is in a plane which is parallel to the yz-plane so the line equation is x x 0 and y y 0 Similarly, when d 2 0, the line equation is y y 0 and d 2 d x x 0,andwhend d 1 d 0, the line equation is z z 0 and x x 0 y y 0 d 1 d 2 Definition: Let two lines L 1 and L 2 be in the direction of d 1 and d 2 Then L 1 and L 2 are parallel if d 1 c d 2 If L 1 and L 2 intersect, thentheangle between L 1 and L 2 is the angle between d 1 and d 2 If L 1 and L 2 are thogonal, then d 1 d 2 0 If L 1 and L 2 are not parallel and not intersecting, then we say they are skew Note that two lines intersect if and only if both lines have a common point Example a Find an equation of the line passing through the point P 1,5,2 and parallel to the vect d,,7 b Determine also where the line intersects the yz plane c Find the line passing through the point P and parallel to the line: x 2 1 y 1 z d Find the line passing through the point P and perpendicular to lines: x 2 1 y 1 z x 1 1t and y 2 2t z 1 t 1

2 a The parametric equations f this line: x 1 t y 5 t z 2 7t b The line intersects with the yz plane when x 0 Then 1 t, t 1,and y 5 1, z Hence the line intersects the yz plane at the point 0, 17, 1 x 1 2t c The direction vect d 2,,1 So the parametric equations of the line: y 5 t z 2 t i j k d The direction vect d 2,,1 1,2, 2 1 7i 5 j 7 k 7,5, The parametric equations of the line is: x 1 7t y 5 5t z 2 7t Example Find the equation of the line through points 1,2, 1 and 5,, d 5 1, 2, 1, 5,5 The parametric equations of the line is x 1 t y 2 5t z 1 5t Example Determine if the lines x 2 t y 1 2t z 5 2t, and x 1 s y 2 s z 1 s are parallel, skew intersect a Check if two lines are parallel: d 1 1,2,2, d 2 1, 1,, sinced 1 cd 2 two lines are not parallel b Check if two lines are intersect (have a common point): Canwefindasand a t such that x,y,z are the same f both parametric equations? Set x :1 s 2 t y :2 s 1 2t z :1 s 5 2t From equation f x we have s 1 t Now substitute s 1 t into equation f y : 1 s 2t we have 1 1 t t 2t So, two lines do not intersect Therefe, two lines are skew 2

2 Planes in R Simple planes: y - a plane parallel to xz-plane and passing the point 0,,0 x 2 - a plane parallel to yz-plane and passing the point 2,0,0 z - a plane parallel to xy-plane and passing

y, z be a point in the plane Then we know the vect P 0 P and the nmal vect n are thogonal, that is, n P 0 P n 1, n 2, n x x 0, y y 0, z z 0 0 n 1 x x 0 n 2 y y 0 n z z 0 0 - the equation f the plane

m 1,m 2, m and n n 1,n 2, n Note that: Two planes are parallel if and only if two nmal vects m and n are parallel, that is: m c n f a constant c Two planes are thogonal if and only if two nmal vects

3 2 Planes in R Simple planes: y - a plane parallel to xz-plane and passing the point 0,,0 x 2 - a plane parallel to yz-plane and passing the point 2,0,0 z - a plane parallel to xy-plane and passing the point 0,0, ; y x 2 z Let P 0 x 0, y 0, z 0 be a point and n n 1,n 2,n beavectinr Find the equation of the plane that contains P 0 and the vect n is nmal to (n is thogonal to the plane) Let P x, y, z be a point in the plane Then we know the vect P 0 P and the nmal vect n are thogonal, that is, n P 0 P n 1, n 2, n x x 0, y y 0, z z 0 0 n 1 x x 0 n 2 y y 0 n z z the equation f the plane n 1 x n 2 y n z n 1 x 0 n 2 y 0 n z 0 n 1 x n 2 y n z n 1 x 0 n 2 y 0 n z 0 0 Clearly, this equation is linear in R Let m 1 x x 0 m 2 y y 0 m z z 0 0andn 1 x x 0 n 2 y y 0 n z z 0 0betwo planes Let m m 1,m 2, m and n n 1,n 2, n Note that: Two planes are parallel if and only if two nmal vects m and n are parallel, that is: m c n f a constant c Two planes are thogonal if and only if two nmal vects m and n are thogonal, that is, m n 0 Example Find an equation of the plane containing the point 1, 2, with nmal vect,5,6 The equation of the plane: x 1 5 y 2 6 z 0 Example Find the plane containing the three points P 1,2,2, Q 2, 1,, and R,5, 2

i j k n PQ PR 1,,2 2,, 1 2 6i 8j 9k 6,8,9 2 The equation of the plane: 6 x 1 8 y 2 9 z 2 0 Example Sketch the place x 2y z 1 x 2y z 1 in the 1st octant x 2y z 1 Example Find an equation f the plane

Find an equation of the line which intersects planes: x 2y z, and x y z 5 Solve x 2y z (1) x y z 5 (2) f xand z in terms of y : (1)-(2):6y 2z 2, y z 1, z y 1 (1): x 2y z 2y y 1 2 5y Since y is free,

4 i j k n PQ PR 1,,2 2,, 1 2 6i 8j 9k 6,8,9 2 The equation of the plane: 6 x 1 8 y 2 9 z 2 0 Example Sketch the place x 2y z 1 x 2y z 1 in the 1st octant x 2y z 1 Example Find an equation f the plane passing through the point 1,, 5 and parallel to the plane defined by 2x 5y 7z 12 The nmal vect of the plane is n 2, 5,7 Then the equation of the plane: 2 x 1 5 y 7 z 5 0, 2x 5y 7z Example Find an equation of the line which intersects planes: x 2y z, and x y z 5 Solve x 2y z (1) x y z 5 (2) f xand z in terms of y : (1)-(2):6y 2z 2, y z 1, z y 1 (1): x 2y z 2y y 1 2 5y Since y is free, let y t The solution is equation x 2 5t y t z 1 t, the parametric equations f the line Example Find an equation of the plane containing the point 1,, 1 and perpendicular to the planes x y 2z 1 and 2x y z 2

5 The nmal vects of the given planes are: n 1 1,1, 2 and n 2 2,1,1 The nmal vect n of the plane is thogonal to both given planes, that is, n n 1 n 2 n i j k i 5j k The equation of the plane is: x 1 5 y z 1 0 Example Find the distance between the parallel planes: 2x y z 6, and x 6y 2z 8 a Pick one point from each plane: P 1 1, 1, 7 from the 1st plane and P 2 0, 0, from the 2nd plane b Find the vect P 1 P 2 : P 1 P 2 1, 1, c Compute comp n P 1 P 2 : n 2,,1 comp n P 1 P 2 P 1P 2 n n

- parametric equations for the line, z z 0 td 3 or if d 1 0, d 2 0andd 3 0, - symmetric equations of the line.

- parametric equations for the line, z z 0 td 3 or if d 1 0, d 2 0andd 3 0, - symmetric equations of the line. Lines and Planes in Space -(105) Questions: 1 What is the equation of a line if we know (1) two points P x 1,y 1,z 1 and Q x 2,y 2,z 2 on the line; (2) a point P x 1,y 1,z 1 on the line and the line is