The vertices of a parallelogram are the origin and the points A(-1,4), B(3,6), and C(7,2). Write the vector equations of the lines that make up the sides of the parallelogram

Are you sure your points are correct?

None of the line segments form parallel lines, (slopes are not equal)

Isn't there another way to find the vector equation because they are the correct coordinates sample answers are:

[x,y]=[-1,4]+t[1,-1]
[x,y]=[-1,4]+t[2,1]

Sure, we can find the vector equations of those lines, but I hesitated since you called it a parallelogram, which it isn't.

let's find vector equation for line BC
direction vector of BC is [4,-4] or simplified to [1,-1)
a point on there is {3,6)

equation for BC:
[x,y] = (3,6) + t(1,-1)

I still think there is something wrong with your question.

To write the vector equations of the lines that make up the sides of the parallelogram, we first need to find the vectors representing these sides.

Let's start with the side formed by the vertices A(-1,4) and B(3,6).

1. Find the vector AB:
To find the vector AB, we subtract the coordinates of point A from the coordinates of point B.

AB = B - A = (3 - (-1), 6 - 4)
= (4, 2)

So, the vector AB is (4, 2).

2. Write the vector equation for the line AB:
To write the vector equation for the line AB, we need a point on the line. We can use point A or point B as the starting point.

Using point A as the starting point, the vector equation of the line AB is:
r = A + t(AB)

where r represents any point on the line and t is a scalar parameter.

Substituting the values, we get:
r = (-1, 4) + t(4, 2)

Now, let's find the vector equation for the side formed by the vertices B(3,6) and C(7,2).

3. Find the vector BC:
We subtract the coordinates of point B from the coordinates of point C.

BC = C - B = (7 - 3, 2 - 6)
= (4, -4)

So, the vector BC is (4, -4).

4. Write the vector equation for the line BC:
Using point B as the starting point, the vector equation of the line BC is:
r = B + t(BC)

Substituting the values, we get:
r = (3, 6) + t(4, -4)

Therefore, the vector equations of the lines that make up the sides of the parallelogram are:
For the line AB: r = (-1, 4) + t(4, 2)
For the line BC: r = (3, 6) + t(4, -4)