Final answer:
To find two parametric representations for the plane through the origin that contains the given vectors, subtract the second vector from the first vector to find the direction vector. Then use this direction vector to form two parametric representations of the plane.
Step-by-step explanation:
To find two parametric representations for the plane through the origin that contains the vectors 4-3 and 3-6, we need to find the direction vectors of the plane. Since the plane contains the origin, we know that the plane must also contain the zero vector (0, 0). Therefore, both of the given vectors lie on the plane. We can use these vectors to find the direction vectors by taking the difference of the given vectors.
Let's call the direction vector a. To find a, subtract the second vector from the first vector:
a = (4-3)i + (3-6)j = i - 3j
Now, we can find two parametric representations for the plane using the equation of a plane:
x = at, y = bt
where a and b are any non-zero real numbers and t is a parameter.
First parametric representation:
x = t, y = -3t
Second parametric representation:
x = 2t, y = -6t