find symmetric equations for the line through the point parallel to the specified line.
(1, -1, 2)
parallel to
x = -7 -4t
y = -5-7t
z = 2- 4t
the line through (1,-1,2) is
(x-1)/a = (y+1)/b = (z-2)/c
and <a,b,c> is the desired direction vector.
Note that your parametric equations can be converted to symmetric by eliminating t:
(x+7)/-4 = (y+5)/-7 = (z-2)/-4
That should be a large enough clue, no?
To find symmetric equations for a line through a point parallel to a given line, we need to determine the direction vector of the given line and use it to write the symmetric equations.
Given line: x = -7 - 4t, y = -5 - 7t, z = 2 - 4t
The direction vector of the given line is (-4, -7, -4).
Since the line we are looking for is parallel to the given line, its direction vector will also be (-4, -7, -4).
Let's denote the point on the line as (x0, y0, z0) = (1, -1, 2).
The symmetric equations for a line with direction vector (a, b, c) passing through (x0, y0, z0) are:
x - x0 / a = y - y0 / b = z - z0 / c
Let's substitute the values into the symmetric equations:
(x - 1) / -4 = (y + 1) / -7 = (z - 2) / -4
Therefore, the symmetric equations for the line through the point (1, -1, 2) parallel to the given line are:
(x - 1) / -4 = (y + 1) / -7 = (z - 2) / -4
To find the symmetric equations of a line through a given point parallel to another line, you can use the vector equation of a line and the given direction vector. Here's how you can do it:
1. Start with the vector equation of a line in terms of a parameter t:
r = <x, y, z> = <x0, y0, z0> + t<d1, d2, d3>
2. Plug in the coordinates of the given point (1, -1, 2) into the vector equation:
<x, y, z> = <1, -1, 2> + t<d1, d2, d3>
3. Determine the direction vector of the line you want to be parallel to. In this case, it is <d1, d2, d3> = <-4, -7, -4>.
4. Substitute the direction vector into the equation from step 2:
<x, y, z> = <1, -1, 2> + t<-4, -7, -4>
5. Distribute the t to each component of the direction vector:
<x, y, z> = <1 - 4t, -1 - 7t, 2 - 4t>
So, the symmetric equations for the line through the point (1, -1, 2) parallel to the line x = -7 - 4t, y = -5 - 7t, z = 2 - 4t are:
x = 1 - 4t
y = -1 - 7t
z = 2 - 4t