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).
First find the midpoint of LM which is P(2,1,-1)
so the direction vector of PQ is [0, 2, -4] or [0,1,-2] in reduced form
so the vector equation is:
(x,y,z) = (2,-1,3) + t(0,1,-2)
You MUST know how to convert this to parametric and symmetric form.
Why did the line go to the therapist?
Because it had a point to make!
To find the vector equation, parametric equations, and symmetric equation for the line passing through Q(2, -1, 3) and the mid-point of the line segment from L(3, -2, 5) to M(1, 4, -7), we can follow these steps:
1. Find the mid-point of the line segment from L to M:
- The mid-point is the average of the coordinates of the endpoints.
- Mid-point coordinates: ((3 + 1)/2, (-2 + 4)/2, (5 - 7)/2) = (2, 1, -1)
2. Find the direction vector for the line passing through Q and the mid-point:
- The direction vector is obtained by subtracting the coordinates of Q from the mid-point.
- Direction vector: (2 - 2, 1 - (-1), -1 - 3) = (0, 2, -4)
3. Vector equation:
- Let P(x, y, z) be any point on the line.
- The vector equation is given by: r = Q + t * (Q_midpoint)
- r = (2, -1, 3) + t * (0, 2, -4)
4. Parametric equations:
- Let x, y, and z be the coordinates of any point on the line.
- The parametric equations are:
x = 2
y = -1 + 2t
z = 3 - 4t
5. Symmetric equation:
- The symmetric equation expresses the relationship between the coordinates of a point on the line.
- The symmetric equation is: (x - 2) / 0 = (y + 1) / 2 = (z - 3) / (-4)
- Note that we divide by the corresponding components of the direction vector.
To summarize:
- Vector equation: r = (2, -1, 3) + t * (0, 2, -4)
- Parametric equations: x = 2, y = -1 + 2t, z = 3 - 4t
- Symmetric equation: (x - 2) / 0 = (y + 1) / 2 = (z - 3) / (-4)
To determine the equations of the line through Q(2, -1, 3) and the midpoint of the line segment LM, we need to find the coordinates of the midpoint first.
Step 1: Find the midpoint of LM
To find the midpoint, we use the formula:
(midpoint) = ( (x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2 )
Let's calculate the midpoint coordinates:
x = (3 + 1)/2 = 4/2 = 2
y = (-2 + 4)/2 = 2/2 = 1
z = (5 - 7)/2 = -2/2 = -1
So, the midpoint of LM is M' (2, 1, -1).
Step 2: Determine the vector equation
The vector equation of a line passing through point Q(2, -1, 3) is given by:
r = r₀ + td
Where:
- r is the position vector of any point on the line.
- r₀ is the position vector of the given point (Q).
- t is a parameter that represents the scale factor.
- d is the direction vector of the line.
To find the direction vector (d), we subtract the coordinates of Q from the coordinates of M':
d = M' - Q = (2, 1, -1) - (2, -1, 3) = (0, 2, -4)
The vector equation of the line is:
r = (2, -1, 3) + t(0, 2, -4)
Step 3: Determine the parametric equations
The parametric equations of the line can be found by expressing the x, y, and z coordinates in terms of the parameter t:
x = 2 + 0t = 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 of the line can be found by expressing the x, y, and z coordinates in terms of ratios:
x - 2 / 0 = y + 1 / 2 = z - 3 / -4
Since the direction vector has a zero component in the x-coordinate, we can simplify the symmetric equations to:
y + 1 / 2 = z - 3 / -4
Thus, the symmetric equations of the line are:
y + 1 = -2t
z - 3 = 4t