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

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

Let's start with the line connecting the points A(-1,4) and B(3,6). The vector equation of this line can be written as:

r = A + t(B - A)

Where r is a position vector on the line, t is a scalar parameter, and A and B are the position vectors of the points A and B, respectively.

Substituting the given values, we have:

r = (-1, 4) + t((3, 6) - (-1, 4))

Simplifying:

r = (-1, 4) + t(4, 2)
r = (-1 + 4t, 4 + 2t)

The vector equation for the line connecting A and B is r = (-1 + 4t, 4 + 2t).

Similarly, we can find the vector equations for the other sides of the parallelogram.

For the line connecting B(3,6) and C(7,2):

r = B + t(C - B)

Substituting the values:

r = (3, 6) + t((7, 2) - (3, 6))

Simplifying:

r = (3, 6) + t(4, -4)
r = (3 + 4t, 6 - 4t)

The vector equation for the line connecting B and C is r = (3 + 4t, 6 - 4t).

For the line connecting C(7,2) and the origin (0,0):

r = C + t(O - C)

Substituting the values:

r = (7, 2) + t((0, 0) - (7, 2))

Simplifying:

r = (7, 2) + t(-7, -2)
r = (7 - 7t, 2 - 2t)

The vector equation for the line connecting C and the origin is r = (7 - 7t, 2 - 2t).

Finally, for the line connecting the origin (0,0) and A(-1,4), the vector equation is simply r = tA, where A is the position vector of point A and t is a scalar parameter:

r = tA
r = t(-1, 4)
r = (-t, 4t)

So, the vector equation for the line connecting the origin and A is r = (-t, 4t).

Therefore, the vector equations of the lines that make up the sides of the parallelogram are:

r1 = (-1 + 4t, 4 + 2t) (connecting A and B)
r2 = (3 + 4t, 6 - 4t) (connecting B and C)
r3 = (7 - 7t, 2 - 2t) (connecting C and the origin)
r4 = (-t, 4t) (connecting the origin and A)

To write the vector equations of the lines that make up the sides of the parallelogram, we need to find the direction vectors for those lines.

Let's start with the line passing through the origin and point A(-1,4). We can find the direction vector by subtracting the coordinates of the origin from those of point A:

Direction vector of line OA = A - O = (-1, 4) - (0, 0) = (-1, 4).

Now, let's move on to the line passing through points A(-1,4) and B(3,6). Again, we find the direction vector by subtracting the coordinates of A from those of B:

Direction vector of line AB = B - A = (3, 6) - (-1, 4) = (3 + 1, 6 - 4) = (4, 2).

Similarly, for the line passing through points B(3,6) and C(7,2):

Direction vector of line BC = C - B = (7, 2) - (3, 6) = (7 - 3, 2 - 6) = (4, -4).

Finally, for the line passing through points C(7,2) and the origin:

Direction vector of line CO = O - C = (0, 0) - (7, 2) = (-7, -2).

Now that we have the direction vectors, we can write the vector equations of the lines using the general vector equation:

For the line OA:

r = (0, 0) + t(-1, 4), where t is a parameter.

For the line AB:

r = (-1, 4) + s(4, 2), where s is a parameter.

For the line BC:

r = (3, 6) + u(4, -4), where u is a parameter.

For the line CO:

r = (7, 2) + v(-7, -2), where v is a parameter.

These equations represent the lines that make up the sides of the parallelogram.