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)

The tangent vectors for L1 and L2 are P' = (3,2,-3) and P' = (-3.5,2,-6.5), respectively. The tangent vector for L1 is constant, but the tangent vector for L2 is also constant, as it is not dependent on any parameter.

In conclusion, both lines have constant tangent vectors.

To find the equations of lines L1 and L2, you can use the given information:

L1: P(u) = P1 + u(P2 - P1)
= (1,3,5) + u(3,2,-3)

L2: P(u) = P4 + v(P3 - P4)
= (2.5,4,3.5) + v(-3.5,2,-6.5)

To find a point on each vector for each line where u = 0.25, you substitute u = 0.25 into the equations:

For L1:
x = 1 + 0.25*3 = 1.75
y = 3 + 0.25*2 = 3.5
z = 5 + 0.25*(-3) = 4.25

So, when u = 0.25 on L1, the point is (1.75, 3.5, 4.25).

For L2:
x = 2.5 + 0.25*(-3.5) = -0.125
y = 4 + 0.25*2 = 5.5
z = 3.5 + 0.25*(-6.5) = -1.375

So, when u = 0.25 on L2, the point is (-0.125, 5.5, -1.375).

To find the tangent vector for each line, you subtract the initial and final points:

For L1:
P' = P2 - P1 = (3,2,-3)

For L2:
P' = P4 - P3 = (-3.5,2,-6.5)

The tangent vector for L1 is (3,2,-3) and for L2 is (-3.5,2,-6.5).

Since the tangent vectors for L1 and L2 are constant, it means that the lines are straight lines. The tangent vectors represent the direction of the lines.

In conclusion, L1 and L2 are straight lines, and their equations, points at u = 0.25, and tangent vectors have been determined.