Write down the equation of the line which passes through the points (2, -1, 3) and (1, 4, -3).  Write all three forms of the equation of the line.

Solution

To do the above task we require the vector v^→ that will be parallel to the line. This can be any type of vector as long as it's parallel to the line.  Generally, v^→ won't lie on the line itself.  Though, in this case it will.

All we require to do is let v^→ be the vector that begins at the second point and ends at the first point. As these two points are on the line the vector among them will as well lie on the line and therefore will be parallel to the line. So,

v^→ = (1, -5, 6)

Note: The order of the points was selected to reduce the number of minus signs in the vector. We could only have simply gone the other way.

Just one time we've got v^→ there really isn't anything else to do. To make use of the vector form we'll require a point on the line.  We have got two and thus we can make use of either one. We'll use the first point. There is a vector form of the line.

r^→ = (2, -1, 3) + t (1, -5, 6) = (2 + t, -1 - 5t, 3+6t)

One time we have this equation the other two forms follow. The following equations are the parametric equations of the line.

x = 2+ t

y = -1 -5t

z = 3 + 6t

The following equation is the symmetric form

(x - 2) / 1

= (y + 1) / -5

= (z - 3) /6