I spent hours on this and I still cant figure it out please help me:
Let L1 be the line passing through the points Q1=(4, 2, −3) and Q2=(5, 5, −5). Find a value of k so the line L2 passing through the point P1 = P1(3, 7, k) with direction vector →d=[−3, −3, 4]T intersects with L1.
L1 is (4,2,-3) + (1,3,-2)t
L2 is (3,7,k) + (-3,-3,4)t
4+t = 3-3n t
2+3t = 7-3n t
4+t-3 = 2+3t-7
t = 3
n = -4/9
Now we can see that
-3-2t = k+4nt
-3-2(3) = k+4(-4/9)(3)
k = -11/3
L2 = (3,7,-11/3) - 4/9 (-3,-3,4)
= (3,7,-11/3) + 4/9 (3,3,-4)
at t=3,
L1(3) = (4,2,-3) + 3(1,3,-2) = (7,11,-9)
L2(3) = (3,7,-11/3) - 4/3 (-3,-3,4) = (7,11,-9)
L2 is (3,7,k) + n(-3,-3,4)t
L2 = (3,7,-11/3) - 4/9 (-3,-3,4)t
= (3,7,-11/3) + 4/9 (3,3,-4)t
To find the intersection of two lines, we need to set up equations for each line and solve them simultaneously. Here's how you can find the value of k in this case:
Step 1: Find the equation of line L1:
The equation of a line passing through two points (Q1 and Q2) can be written as:
x = x1 + t(x2 - x1)
y = y1 + t(y2 - y1)
z = z1 + t(z2 - z1)
Plugging in the coordinates of Q1 and Q2, we get:
x = 4 + t(5 - 4) -> x = 4 + t
y = 2 + t(5 - 2) -> y = 2 + 3t
z = -3 + t(-5 - (-3)) -> z = -3 - 2t
So, the equation of line L1 becomes:
L1: x = 4 + t, y = 2 + 3t, z = -3 - 2t
Step 2: Find the equation of line L2:
The equation of a line passing through a point (P1) with a direction vector (d) can be written as:
x = x1 + td1
y = y1 + td2
z = z1 + td3
Here, P1 is given as (3, 7, k) and the direction vector is given as (-3, -3, 4), so the equation of line L2 becomes:
L2: x = 3 - 3t, y = 7 - 3t, z = k + 4t
Step 3: Solve the equations simultaneously:
By equating the x, y, and z coordinates of L1 and L2, we can solve for t and k:
x-coordinate: 4 + t = 3 - 3t
Simplifying, we get: 4t = -1 -> t = -1/4
y-coordinate: 2 + 3t = 7 - 3t
Simplifying, we get: 6t = 5 -> t = 5/6
z-coordinate: -3 - 2t = k + 4t
Simplifying, we get: 6t + 3 = k + 4t -> k = 6t + 3 - 4t -> k = 2t + 3
Substituting the value of t into the equation for k, we get:
k = 2 * (5/6) + 3
k = 10/6 + 3
k = 10/6 + 18/6
k = 28/6
k = 14/3
So, the value of k that makes line L2 intersect with line L1 is k = 14/3 or k ≈ 4.667.