Vector Help?

Question

The equation X (t ) = A + t L is the parametric
equation of a line through the point P : (2, −3, 1) .
The parameter t represents distance from the point
P, directed so that the I component of L is positive.
We know that the line is orthogonal to the plane with
equation 1x + 3y − 2z = −3. Then
A= ___I+ ___J+ ___K
L= ___I+ ___J+ ___K

I know that A= 2I + -3J +2K, but how do I find L? And it isn't 1, 3,-2.

Answer 1

You can write the equation of a line

X(t) = A + tL
where
A is a located vector from the origin to point A
L is the directional vector of the line
t is a scalar that ranges over the real numbers

Let A = OP = <2, -3, 1>

The line is orthogonal to the plane:

x + 3y - 2z = -3

This plane has the normal vector n = <1, 3, -2>. This is also the directional vector of the line since the line is orthogonal to the plane.

The equation of the line is:

r(t) = OP + tn
r(t) = <2, -3, 1> + t<1, 3, -2>

A = OP = <2, -3, 1> = 2i - 3j + k
L = n = <1, 3, -2> = i + 3j - 2k

If you want L to be normalized, divide by its magnitude.

L / || L || = L / √14

But this is not necessary to write the equation of the line.

Answer 2

The answer is:
A = 2I - 3J +1K
L = (1I +3J - 2K) / sqrt(14)

So... 1,3,-2 was "almost" correct for L. The only thing missing is the factor of 1 / sqrt(14). That factor is needed because the problem says that t represents the distance from the point P. That implies that L has to be a unit vector. (a vector whose length is 1). Multiplying by 1 / sqrt(14) gives us a vector in the same direction as (1,3,-2), but with a length of 1.

2I - 3J + 2K is not a correct answer for A. A has to be equal to P, because the problem says that the line must pass through P. So, A = P = 2I -3J + 1K

Answer 3

I'm not good in English but I think there is a mistake in question. When you're in 3 dimensional space you need to express a line equation with two "=" sign like (x-d)/a=(y-e)/b=(z-f)/c
and when you express an equation like the one you do, it means a plane. ( sorry for my English)
If I misunderstood the question please explain more, I believe I can help, thanks.
You can IM me if you want