Determine vector, parametric and, if possible, symmetric equations for the line through Q(2, -1, 3) and the mid-point of the line segment from L(3, -2, 5) to M(1, 4, -7). Not sure what to do with the mid-point.
To find the vector, parametric, and symmetric equations for the line, let's break it down into steps:
Step 1: Find the midpoint of the line segment L(3, -2, 5) to M(1, 4, -7).
The midpoint formula is given by:
Midpoint (Mx, My, Mz) = ((Lx + Mx) / 2, (Ly + My) / 2, (Lz + Mz) / 2)
Substituting the coordinates, we get:
Mx = (3 + 1) / 2 = 2
My = (-2 + 4) / 2 = 1
Mz = (5 - 7) / 2 = -1
So, the midpoint is M(2, 1, -1).
Step 2: Determine the vector equation.
The vector equation of a line passing through point Q(2, -1, 3) with direction vector v = <a, b, c> is given by:
r = Q + tv
where r is the position vector of any point on the line, and t is a scalar parameter.
To determine the direction vector, we need to find the displacement vector between Q and the midpoint M:
v = M - Q = <Mx - Qx, My - Qy, Mz - Qz> = <2 - 2, 1 - (-1), -1 - 3> = <0, 2, -4>
The vector equation becomes:
r = <2, -1, 3> + t<0, 2, -4>
Simplifying, we get:
x = 2
y = -1 + 2t
z = 3 - 4t
So, the vector equation of the line is:
r = <2, -1, 3> + t<0, 2, -4> or
r = <2, -1, 3> + t<0, 1, -2>
Step 3: Determine the parametric equations.
The parametric equations are obtained by separating each component of the vector equation.
x = 2
y = -1 + 2t
z = 3 - 4t
So, the parametric equations of the line are:
x = 2
y = -1 + 2t
z = 3 - 4t
Step 4: Determine the symmetric equations.
The symmetric equations are obtained by isolating the parameter t in each parametric equation.
From y = -1 + 2t, we get t = (y + 1) / 2. Then substituting this into the other equations, we get:
x = 2
z = 3 - 4(y + 1) / 2 = 3 - 2(y + 1) = 3 - 2y - 2
Simplifying further, we get:
x = 2
2y + z = 1
So, the symmetric equations of the line are:
x = 2
2y + z = 1
To determine the vector, parametric, and symmetric equations for the line, we can follow these steps:
1. Find the midpoint of the line segment from L to M.
2. Determine the direction vector for the line.
3. Use the information from steps 1 and 2 to write the vector, parametric, and symmetric equations for the line.
Let's go through each step in detail:
1. Finding the midpoint:
The midpoint of the line segment from L(3, -2, 5) to M(1, 4, -7) can be found by taking the average of the coordinates of the two points. We add the corresponding coordinates and divide by 2:
Midpoint = [(3 + 1) / 2, (-2 + 4) / 2, (5 + (-7)) / 2]
= [2 / 2, 2 / 2, -2 / 2]
= [1, 1, -1]
So, the midpoint is Q(1, 1, -1).
2. Determining the direction vector:
The direction vector of the line can be obtained by subtracting the coordinates of the two given points. Let's subtract the coordinates of Q(2, -1, 3) from the coordinates of the midpoint Q(1, 1, -1):
Direction vector = [1 - 2, 1 - (-1), -1 - 3]
= [-1, 2, -4]
Hence, the direction vector of the line is [-1, 2, -4].
3. Writing the equations for the line:
a) Vector equation:
A vector equation can be written as:
r = Q + t * d,
where r is the position vector of any point on the line, Q(2, -1, 3) is a known point on the line, t is a parameter, and d[-1, 2, -4] is the direction vector.
So, the vector equation for the line is:
r = [2, -1, 3] + t * [-1, 2, -4].
b) Parametric equations:
The parametric equations can be obtained by separating the x, y, and z components:
x = 2 - t,
y = -1 + 2t,
z = 3 - 4t.
c) Symmetric equations:
The symmetric equations can be written by isolating the parameter t in the parametric equations:
t = 2 - x,
t = (y + 1) / 2,
t = (3 - z) / 4.
These are the symmetric equations for the line.
To summarize, the vector equation for the line is r = [2, -1, 3] + t * [-1, 2, -4], the parametric equations are x = 2 - t, y = -1 + 2t, z = 3 - 4t, and the symmetric equations are t = 2 - x, t = (y + 1) / 2, t = (3 - z) / 4.