State a direction vector with integer components.
x = (1/3) + 2t, y = 3-(2/3)t
State the integer components and name a point on the line with integer coordinates.
r = (1/3, 1/2) + t(1/3, 1/4)
Please explain.
To find a direction vector with integer components, we can rewrite the given line equation in the form of vector equation by isolating the x and y terms.
Given:
x = (1/3) + 2t
y = 3 - (2/3)t
Using the vector equation format, we have:
r = (x, y) = ((1/3) + 2t, 3 - (2/3)t)
Now, we can write the direction vector by considering the coefficients of 't' for x and y:
v = (2, -2/3)
This vector v is a direction vector; it represents the direction of the line at any point.
To find a point on the line with integer coordinates, we can substitute integer values for t and solve for x and y.
For example, let's choose t = 1:
x = (1/3) + 2(1) = (1/3) + 2/1 = 1 + 2/3 = 1 2/3 = 5/3
y = 3 - (2/3)(1) = 3 - 2/3 = 9/3 - 2/3 = 7/3
So, when t = 1, we have the point (5/3, 7/3). This point lies on the line and has integer coordinates.
Therefore, we can say that the point (5/3, 7/3) is on the line with integer coordinates.