L1=P1(1,3,5) and P2(4,5,2)
L2=P4(-1,6,-3) and P3(points not given)
a. Find the equations of lines L1 and L2. P3 is the midpoint of L1, that is, u=0.5 at P3.
P(u)=P1 + u((P2-P1) 0 which is less than or equal to (u)which is less than or equal to 1.
L1= P(u)= (1,3,5) + u(3,2,-3)
L2= P(u)= (2.5,4,3.5)+ v(-3.5,2,-6.5)
b. Find a point on each vector for each line where u=0.25.
L1
2.5 +1= (3.2/2)=1.75
4 + 3 = (7/2)= 3.5
3.5 + 5= (8.5/2)=4.25
L2
0.75+(-1)= (-0.25/2)= -0.125
5+ 6= (11/2)= 5.5
0.25 + (-3)= (-2.75/2) = -1.375
ans= when u =0.25 on L1 the points are (1.75,3.5,4.25). When u=0.25 on L2 the points are (-0.125,5.5,-1.375).
c. Find the tangent vector for each line. Are they constant? What is your conclusion.
L1
P`= P2-P1
P`= (3,2,-3)
L2
P`= P4-P3
P`= (-3.5,2,-6.5)
To find the equations of lines L1 and L2, we use the formula P(u) = P1 + u(P2 - P1), where P(u) represents a point on the line, P1 and P2 are the given points that determine the line's direction, and u is a scalar variable. For L1, we substitute the given values:
L1: P(u) = (1,3,5) + u(3,2,-3)
For L2, we are also given a midpoint P3, which means u = 0.5 at P3. Using this information, we can find the direction vector (P4 - P3):
L2: P(u) = (2.5,4,3.5) + v(-3.5,2,-6.5)
To find a point on each line where u = 0.25, we substitute u = 0.25 into the equations:
For L1:
x-coordinate: 1 + 0.25 * 3 = 1.75
y-coordinate: 3 + 0.25 * 2 = 3.5
z-coordinate: 5 + 0.25 * -3 = 4.25
Therefore, when u = 0.25 on L1, the coordinates are (1.75, 3.5, 4.25).
For L2:
x-coordinate: 2.5 + 0.25 * -3.5 = -0.125
y-coordinate: 4 + 0.25 * 2 = 5.5
z-coordinate: 3.5 + 0.25 * -6.5 = -1.375
Therefore, when u = 0.25 on L2, the coordinates are (-0.125, 5.5, -1.375).
To find the tangent vector for each line, we differentiate the equations of the lines with respect to u:
For L1:
P' = (3, 2, -3)
For L2:
P' = (-3.5, 2, -6.5)
The tangent vectors for both lines are constant, meaning they do not change as u varies.
In conclusion, the equations of lines L1 and L2 are given by:
L1: P(u) = (1,3,5) + u(3,2,-3)
L2: P(u) = (2.5,4,3.5) + v(-3.5,2,-6.5)
The points on each line when u = 0.25 are:
L1: (1.75, 3.5, 4.25)
L2: (-0.125, 5.5, -1.375)
The tangent vectors for each line are constant, indicating that the lines do not change direction as u varies.