4. Write a vector equation of the line through the point (5, -2, 3) and parallel to the vector v=[4, -3, 1]
You know the direction of the line has to be [4,-3,1] , so
r = (5,-23) + t(4,-3,1)
(how easy was that?)
thank you reiny, that's the answer i first got but i wasn't sure if it was right, thanks
To find the vector equation of a line parallel to a given vector and passing through a given point, we can use the parametric equation of a line. The parametric equation is given by:
P = P0 + t * v
where P is a point on the line, P0 is a known point on the line, v is the direction vector, and t is a parameter that determines different points on the line.
In this case, the known point on the line is (5, -2, 3), which we'll call P0, and the direction vector is v = [4, -3, 1].
So, the vector equation of the line can be written as:
P = (5, -2, 3) + t * [4, -3, 1]
or, in component form:
x = 5 + 4t
y = -2 - 3t
z = 3 + t
This equation represents a line passing through the point (5, -2, 3) and parallel to the vector v = [4, -3, 1].