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 can use the formula P(u) = P1 + u(P2 - P1), where P(u) represents a point on the line, P1 and P2 are given points on the line, and u is a scalar ranging from 0 to 1.

a. For L1:
P1 = (1, 3, 5)
P2 = (4, 5, 2)
Using the formula, we can substitute these values into P(u) = P1 + u(P2 - P1):
L1 = P(u) = (1, 3, 5) + u(3, 2, -3)

For L2:
P4 = (-1, 6, -3)
P3 (midpoint) is not given, but we know that it is the midpoint of L1. So, we can find P3 by using the formula P3 = P1 + 0.5(P2 - P1):
P3 = (1, 3, 5) + 0.5(3, 2, -3) = (2.5, 4, 3.5)
Now, using P3, we can find L2:
L2 = P(u) = (2.5, 4, 3.5) + v(-3.5, 2, -6.5)

b. To find a point on each line where u = 0.25, we substitute u = 0.25 into the equations of L1 and L2:
For L1:
P(u) = (1, 3, 5) + 0.25(3, 2, -3) = (1.75, 3.5, 4.25)
For L2:
P(u) = (2.5, 4, 3.5) + 0.25(-3.5, 2, -6.5) = (-0.125, 5.5, -1.375)

So, when u = 0.25, the points on L1 are (1.75, 3.5, 4.25) and on L2 are (-0.125, 5.5, -1.375).

c. To find the tangent vector for each line, we take the derivative of the equation of the line with respect to u. The tangent vector represents the direction the line is moving in.
For L1: P' = P2 - P1 = (3, 2, -3)
For L2: P' = P4 - P3 = (-3.5, 2, -6.5)

The tangent vectors for both lines are not constant since they depend on the values of P1, P2, P3, and P4.

In conclusion, the equations of the lines L1 and L2 are found using the point-slope form. The points on each line where u = 0.25 are determined by substituting the value of u into the respective equations. The tangent vectors for each line represent the direction the line is moving in, and they are not constant.