Determine the vector parametric equation of the line through the origin
and the intersection of the lines (find the co-ordinates of this intersec-
tion as well)
v(t) = { 2t + 3
−4t
t−3
}
w(s)
{
10s + 3
5s − 25
4 − 2s}
To determine the vector parametric equation of the line through the origin and the intersection of the given lines, we first need to find the coordinates of their intersection.
To find the intersection point of the lines defined by the equations v(t) and w(s), we need to set the corresponding components equal to each other and solve for t and s.
Setting the x-components equal to each other:
2t + 3 = 10s + 3
Simplifying, we get:
2t = 10s
t = 5s
Setting the y-components equal to each other:
-4t = 5s - 25
-4(5s) = 5s - 25
-20s = 5s - 25
25s = 25
s = 1
Substituting s = 1 into t = 5s, we get t = 5(1) = 5.
Therefore, the intersection point is (t, s) = (5, 1).
Now we can construct the vector parametric equation of the line passing through the origin and the intersection point.
The parametric equation of a line passing through the origin is given by r(t) = tv, where v represents the direction vector of the line.
The direction vector v can be obtained by subtracting the coordinates of the origin from the coordinates of the intersection point. In other words:
v = (5, 1) - (0, 0)
v = (5, 1)
Therefore, the vector parametric equation of the line passing through the origin and the intersection point is:
r(t) = t(5, 1)
This equation represents all the points on the line, where t is a scalar parameter.