1. Rubono Setiawan, S.Si.,M.Sc.
Calculus III
Second Lecture Notes By
Rubono Setiawan, S.Si.,M.Sc.
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2. Contents of this presentation
1.2. Equations of Lines and Curve

3. Lines
A line in the xy-plane is determined when a point on the line and the direction of the line ( its slope or angle of inclination) are given. The equation of the line can then be written using the point-slope form.
Likewise, a line L in three- dimensional space is determined when we know a point Po(X0,Y0,Z0) on L and the direction of L. In three dimensional space the direction of line is conveniently described by a vector, so we let v be a vector parallel to L. Let P(x,y,z) be an arbitrary point on L and let r0
and r be the position vectors of P0 and P
( that is, they have representations and )

4. If a is a vector with representation then the Triangle Law for vector addition gives r = r0 + a. But, since a and v are parallel vectors, there is a scalar t such that a = tv. Thus
r = r0 + tv
which is a vector equation of L .
Each value of parameter t gives the position vector r gives the position vector r of a point on L. In other words, as t varies, the line is traced out by the tip of the vector r. As Following figure, positive values of t correspond to points on L that lie on one side of P0 , whereas negative values of t correspond to points on L that lie on the other side of P0

5. <x,y,z>= < x0+ta, y0+tb, z0+tc >
If the vector v that gives the direction of the line L is written in component form as v = < a, b, c > , then of course, we have
tv=< ta, tb, tc > .We also can write r = <x,y,z> and r0 =< x0,y0,z0>
so the vector position of P (x, y, z) becomes
<x,y,z>= < x0+ta, y0+tb, z0+tc >

6. Two vectors are equal if and only if corresponding components are equal.
Therefore, we have the scalar equations
x = x0+ at y = y0 + bt z= z0+ct *)
where Equations *) are called parametric equations of the line L through the point Po(X0,Y0,Z0) and parallel to the vector v = <a,b,c>. Each value of parameter t gives point (x,y,z) on L

7. Example 1
1. Find a vector equation and parametric equation for the line that passes through the point ( 5,1,3) and its parallel to the vector i + 4j – 2k . Then, find two other points on the line.
2. Find a vector equation and parametric equations for the line through the point (-2,4,10) and parallel to the vector < 3,1,-8>

8. In general , if a vector v = <a,b,c> is used to describe the direction of a line L, then the numbers a, b, and c are called direction numbers of L. Since any vector parallel to v could also be use, we see that any three numbers proportional to a, b, c could also be used as a set of direction numbers of L
Another way to describing a line L is to eliminate the parameter t from Equation *). If none of a, b, or c is 0, we can solve each equation for t, equate the results, and obtain :

9. These equations are called symmetric equations.

10. Example 2
1. a. Find the parametric equations and symmetric equations of the line that passes through the points A (2,4,-3) and B ( 3,-1,1)
b. At what point does this line intersect the xy – plane ?

11. SKEW LINES
In general

12. SKEW LINES
Two lines with certain parametric equations are called by skew lines, if they do not intersect and are not parallel ( and therefore do not lie in same plane )
Example
Show that the lines L1 and L2, with parametric equations :
are skew lines !
Solution
The lines are not parallel, because the corresponding vectors < 1,3,-1> and <2,1,4> are not parallel ( Their components are not proportional ).

13. SKEW LINES
If L1 and L2 had a point of intersection, there would be values of t and s, such that
1+t = 2s
-2+3t=3+s
4-t=-3+4s
But if we solve the first two equations, we get t=11/5 and s=8/5, and these values don’t satisfy the third equation. Therefore, there are no values of t and s that satisfy the three equations. Thus, L1 and L2 does not intersect. Hence, L1 and L2 are skew lines.

15. 1.1.Equations of Lines and Curve ----Curve------
Suppose that f, g, h are continous real function real valued functions on an interval I. Then the set C of all point (x,y,z) in a space, where
x= f(t) y = g(t) z= h(t) (1)
and t is a varies throughout the interval I, is called a Space curve .
The equations in (1) are called parametric equations of C and t is called a parameter.
We can think of C as being traced out by moving particle whose position at time t is (f (t ), g (t ), h (t) ).
If we now consider the vector position
r (t)=< f(t),g(t),h(t)> or r(t)=f(t)i+g(t)j+h(t)k

16. Then r(t) is the position vector of the point P(f(t),g(t),h(t) )on C
Then r(t) is the position vector of the point P(f(t),g(t),h(t) )on C . Thus, any continuous vector function r defines a space curve C that is traced out by the tip of the moving vector r(t) , as shown in Following figure

17. Plane curves also represented in vector notation
Plane curves also represented in vector notation. For instance, the curve given by the parametric equations and
could also described by the vector equation :
where i = <1,0> and j=<0,1>.
Let’s we see the following example

18. Example 3 Describe the curve defined by vector function :
r (t) = < 1+t , 2+5t, t >
Sketch the curve whose vector equation is
r (t ) = cos t i + sin t j + t k
3. Find a vector equation and parametric equations for the line segment that joins the point P(1,3,-2) to the point Q(2.- 1,3).