0.5 Lines and Planes in Space Study Book p 20, Grossman 3.5, Stewart 9.5

1. 0.5 Lines and Planes in SpaceStudy Book p 20, Grossman 3.5, Stewart 9.5 The equation of a LINE in spaceR3 looks quite different from the forms we generally use in the planeR2 : y = mx + k and ax + by = c Given a point P on the line, and the direction of the line (as a direction vector, rather than a slope) we can use vectors to get to any point on the line as follows:

2. Example: Suppose line L passes through points P, Q. . Origin OQ(2,3,5) P(1,0,2) R(x,y,z) To get the coords of Rhalfway between P & Q, we must travel vector OR. One way is to travel OP, then half of vector PQ: ie OR = OP + 1/2 PQ Vector PQ = (2, 3, 5) - (1, 0, 2) = (1, 3, 3). Hence point R(x, y, z) is given by (x, y, z) = (1, 0, 2) + (1/2) (1, 3, 3)

3. To get from origin O to Q, we can travel vector OP, then add all of vector PQ (not just half): ie OQ = OP + 1 PQ gives the coords of Q. If we add a bigger multiple of PQ , we get to a point S beyond Q: eg OP + 2 PQ . If we add a negative multiple of PQ, we get to a point T which lies before P. eg OP - 2PQ . If we add all possible multiples of PQ, ieOP + t PQ t any scalar, we get all possible points on line L.

4. In general: we don’t need to know 2 points on the line. Knowing any 1 point on the line & a direction vector will do. A direction vector for the line is any vector with the same direction as the line, ie parallel to it. Suppose L passes through P(x1, y1, z1) in a direction parallel to vectorv = (a,b,c) . OR(x,y,z) Pv To reach any point R(x, y, z) from the origin, we can travel OP first, andthen a suitable multiple of v: ie OR = OP + t v , t some scalar.

5. We can get all points on L by using all possible values of parameter t. Hence one way to describe the coordinates of any point on the line through P in a direction parallel to v is OR = OP + t v , t any real number. We call this the vector equation of the line. Substituting coordinates gives (x, y, z) = (x1, y1, z1) + t (a, b, c) If weuse just an interval of valuesfor t we getonly part of the line.

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7. Separating x, y & z components in the form (x, y, z) = (x1, y1, zp) + t (a, b, c) gives the Parametric Form of the eqn of the line: x = x1, + t a y = y1, + t b z = zp + t c Notice where the direction vector (a,b,c) is now! And where the known point on the line (x1, y1, zp) is.

8. If we solve the 3 parametric eqns for t & equate them, we get the Symmetric Form of the equation of the line: Notice where the direction vector is now. Note in Ex 3, p 195: any component that would result in division by 0 is listed separately, rather!

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10. Homework on equations for lines Appendix A Problems 3.5: • Master 1 - 17, 22 - 25. • Write full solutions to Q 3, 7, 13, 15 & 17.

11. Objectives Be able to • find the vector equation of a line given 2 points • find the vector equation given 1 point & a direction vector • find the equation in parametric & symmetric form too • identify a direction vector for the line from any of the 3 forms • find points on the line, given its equation in any form