Lines and Planes in Space

Lines and Planes in Space - Lines and Planes in Space(10.5...

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Lines and Planes in Space - (10.5) 1. Lines A line passes through the point P 0 x 0 , y 0 , z 0 in the direction d ! "# d 1 , d 2 , d 3 $ . Let P ! x , y , z " be a point on the line. Then the vector P 0 P " td ! where t is a scalar. Hence, P 0 P "# x ! x 0 , y ! y 0 , z ! z 0 $" td ! "# td 1 , td 2 , td 3 $ or x ! x 0 " td 1 y ! y 0 " td 2 z ! z 0 " td 3 - parametric equations for the line, or x ! x 0 d 1 " y ! y 0 d 2 " z ! z 0 d 3 ! symmetric equations of the line. 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 " cd ! 2 . ! If L 1 and L 2 intersect, then the angle between L 1 and L 2 is the angle between d ! 1 and d ! 2 . ! If L 1 and L 2 are orthogonal, then d ! 1 # d ! 2 " 0. ! If L 1 and L 2 are not parallel and not intersecting, then we say they are skew. Example Find an equation of the line through the point ! 1,5,2 " and parallel to the vector d ! "# 4,3,7 $ . Determine also where the line intersects the yz ! plane. The parametric equations for this line: x ! 1 " 4 t y ! 5 " 3 t z ! 2 " 7 t . The line intersects with the yz ! plane when x " 0. Then ! 1 " 4 t , or t " !
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