Determine vector and parametric equations for the line through the point (2, 1) and parallel to the line with equation (x, y) = (3, -8) + t(-3, -2).

For your given vector equation (x, y) = (3, -8) + t(-3, -2)

the parametric equation would be
x = 3 - 3t
y = -8 - 2t

Form the new equation. Keep in mind that, since it is parallel to the above, the direction must be the same

To determine the vector and parametric equations for the line through the point (2, 1) and parallel to the line with equation (x, y) = (3, -8) + t(-3, -2), we need to find a vector that is parallel to the given line.

The given line is in parametric form: (x, y) = (3, -8) + t(-3, -2), where (3, -8) represents a point on the line, and (-3, -2) represents the direction vector.

To find a vector parallel to the line, we can take the direction vector (-3, -2) of the given line. Since any scalar multiple of a vector gives a parallel vector, we can use this vector as our direction vector.

Therefore, the vector parallel to the line is (-3, -2).

Now, we can use this parallel vector to determine the vector and parametric equations for the line through the point (2, 1).

The vector equation of a line passing through point P(x₁, y₁) and parallel to vector v(a, b) is given by:

R = P + tv,

where R(x, y) represents any point on the line, t is a parameter, P(x₁, y₁) is the given point on the line, and v(a, b) is the parallel vector.

In our case, P(2, 1) is the given point, and v(-3, -2) is the parallel vector we found earlier.

Substituting these values into the equation, we get:

R = (2, 1) + t(-3, -2).

This represents the vector equation of the line passing through the point (2, 1) and parallel to the line with equation (x, y) = (3, -8) + t(-3, -2).

To convert this vector equation into parametric form, we can separate the x and y components as follows:

x = 2 - 3t,
y = 1 - 2t.

These are the parametric equations for the line passing through the point (2, 1) and parallel to the given line.