4. Write a vector equation of the line through the point (5, -2, 3) and parallel to the vector v=[4, -3, 1]

The line would look very much like the one shown in the question:

http://www.jiskha.com/display.cgi?id=1337962346
You only have to replace the point and the vector.

how do i do that...

is it like this

r = (5, -2, 3) + t (4, -3, 1)

thats all i have to do for this question?

Yep, that's it.

To write a vector equation of the line through a given point and parallel to a given vector, we can use the following equation:

r = r₀ + tv

where:
- r is the position vector of any point on the line,
- r₀ is the position vector of the given point on the line (in this case, (5, -2, 3)),
- t is a scalar parameter, and
- v is the given vector parallel to the line.

Given that the point is (5, -2, 3) and the vector is v = [4, -3, 1], we can substitute these values into the equation to get the vector equation of the line:

r = [5, -2, 3] + t[4, -3, 1]

Thus, the vector equation of the line through the point (5, -2, 3) and parallel to the vector v = [4, -3, 1] is r = [5 + 4t, -2 - 3t, 3 + t].