If the line l has parametric equations x=5-3t, y=-2+t, z=1+9t, find the parametric equations for the line through p(5,4,-3) that is parallel to l.
To find the parametric equations for the line through point P(5, 4, -3) that is parallel to line l, we need to use the direction vector of line l to determine the direction of the new line.
Since line l has parametric equations x = 5 - 3t, y = -2 + t, and z = 1 + 9t, we can see that the direction vector of line l is given by the coefficients of t in each equation. In this case, the direction vector is (x, y, z) = (-3, 1, 9).
Now, we can use this direction vector to find the parametric equations for the new line:
x = x0 + a(-3)
y = y0 + a(1)
z = z0 + a(9)
where (x0, y0, z0) is the given point (5, 4, -3) and "a" is a parameter.
Plugging in the values, we get:
x = 5 + (-3a)
y = 4 + a
z = -3 + 9a
Therefore, the parametric equations for the line through P(5, 4, -3) parallel to line l are:
x = 5 - 3a
y = 4 + a
z = -3 + 9a, where "a" is a parameter.
To find the parametric equations for the line through point P(5, 4, -3) that is parallel to line l, we need to determine the direction vector of line l and then use it to construct the equations for the new line.
The direction vector of line l can be obtained by taking the coefficients of t in the parametric equations:
l: x = 5 - 3t, y = -2 + t, z = 1 + 9t
The direction vector of line l is given by the coefficients of t as coefficients of i, j, and k:
l_direction = (-3, 1, 9)
Since the line we want is parallel to line l, it will have the same direction vector. Therefore, the parametric equations for the new line can be written as:
x = 5 + (-3t) = 5 - 3t,
y = 4 + t = 4 + t,
z = -3 + 9t
So, the parametric equations for the line through P(5, 4, -3) that is parallel to line l are:
x = 5 - 3t,
y = 4 + t,
z = -3 + 9t