direction of a line?

Question

if i have 2 points, in this case (1,2,6) and (-2,-1,3) and a line passing through them, how do i find the direction this line is going?

to the first answer:
i dont have vectors do i? i have two points.
plus, which one would i subtract from which?

Answer 1

The easiest way to express a direction is with a unit vector. Take the difference of the two vectors. Then divide that by the magnitude of the difference. That will give you a unit vector in the direction of the line.

You can use that unit vector to get angles if you want.

colattitude = arc cos (z)

the other angle is arc tan (y/x)

Answer 2

In three dimensions it is easiest to calculate the directional vector for the line. This serves the same purpose as the slope of the line does in two dimensions.

Given the points P(1,2,6) and Q(-2,-1,3) calculate the directional vector v, of the line.

v = PQ = Q - P = <-2-1, -1-2, 3-6> = <-3, -3, -3>

Any non-zero multiple of v is also a directional vector of the line. Divide by -3 to simplify the math.

v = <1, 1, 1>

v is the direction of the line.

You can now write the equation of the line L, with the directional vector v and a point. Let's choose P.

L = P + tv
L = <1, 2, 6> + t<1, 1, 1>
where t is a scalar ranging over the real numbers

Please note, if we had used v = <-3, -3, -3> it would still be the same line, but we would reach the points on it with different values of t.