A line has the same x-intercept as
[x,y,z]=[-21,8,14]+t[-12,4,7] and the same y-intercept as
[x,y,z]=[6,-8,12]+s[2,-5,4]. Write the parametric equation of the line.
As I understand it, the x-intercept of a line in 3D would be where y=0 and z=0
that is,
[x,0,0] = [-21,8,14] + t[-12,4,7]
this results in 2 different values of t, thus the line misses the x-axis.
The same would be true for the y-intercept of the 2nd line, it would miss the y-axis.
Writing the parametric equations of the given line one
x=-21-12t and since we want the x intercept the y=z=0 therefore
8+4t=0 and 14+7t=0 in both case the t value is '-2' repeating the similar procedure to find the y-intercept the value of s=-3
there after how do i find my parametric equations.
You are correct, I made an arithmetic error trying to do it in my head.
ok, so when t = -2
[x,y,z] = [3,0,0] , so the endpoint of that vector is(3,0,0,)
and in the second, when s = -3
[x,y,z] = [0, 7,0] so the endpoint of that vector is (0,7,0)
so the direction vector of the line joining the intercepts is (3,-7,0)
and the equation of that line is
[x,y,z] = (3,0,0) + k(3,-7,0)
To find the parametric equation of a line with the same x-intercept and y-intercept as the given lines, we need to determine the values of t and s that satisfy these conditions.
For the line [x, y, z] = [-21, 8, 14] + t[-12, 4, 7], the x-intercept occurs when y and z are both zero. So, we can set y = z = 0 and solve for t.
0 = 8 + 4t => 4t = -8 => t = -2
Similarly, for the line [x, y, z] = [6, -8, 12] + s[2, -5, 4], the y-intercept occurs when x and z are both zero. Setting x = z = 0 and solving for s gives us:
0 = 6 - 2s => 2s = 6 => s = 3
Now, we have the values of t and s that correspond to the x-intercept and y-intercept, respectively.
Therefore, the parametric equation of the line is:
[x, y, z] = [-21, 8, 14] + (-2)[-12, 4, 7] + (3)[2, -5, 4]
Simplifying this equation, we get:
[x, y, z] = [-21 + 24 + 6, 8 - 8 - 15, 14 - 14 + 12]
[x, y, z] = [9, -15, 12]
So, the parametric equation of the line is:
x = 9
y = -15
z = 12