Vectors and Plane: Line reflected in the vector question

Given that

$l_1: \mathbf{r} = (6\mathbf{i} + 2\mathbf{j} -2\mathbf{k}) + \lambda (4\mathbf{i}+5\mathbf{j} -1\mathbf{k})$

and

$\displaystyle \Pi : \mathbf{r} . (2\mathbf{i} -1\mathbf{j} + 4\mathbf{k}) = 4$

We are told that $l_1$ is reflected in the plane $\displaystyle \pi$

And that the reflected line is $l_2$

work out $l_2$

So I started by drawing a sketch which looked like this:

I worked out the value of A, and i got it as $\mathbf{A} = (-2 \mathbf{i} -8 \mathbf{j})$



Then I don't know what to do. I want to draw a perpendicular line through the plane, which would hit both $l_1$ and $l_2$ at equal distance from the plane, but still don't feel like that's enough to find C.

Because if I can work out C, and since I have A, I can then work out a vector formula for $l_2$

How would I go about going from there?

You need 2 points to define a line. You got A, now you can take any point you want on $\ell_1$ and reflect it in the plane to get another.

How would i go about reflecting a point on a plane? I don't think I know how to do that...

Find the perpendicular vector from your point to the plane, and multiply it by 2, and add it onto your point.

I let $\lambda=1$ and I got another value on $\displaystyle l_1$.

This point I named B and I got it to be $(10\mathbf{i} + 7\mathbf{j} - 3\mathbf{k})$

So I would have to reflect this value on the plane by working out the direction vector from A to B, which is AB, and I found AB.

What would I do on from here? Since u told me to find a perpendicular vector, I would need another vector since u have many perpendicular lines to a single line.

Seems to me like you misunderstood, so I'll break it down in steps.

(a) You know the perpendicular vector to the plane, so you can construct a perpendicular line $\ell$ to the plane through this point with a parameter $t$.

(b) then determine for which value of $t$ this line intersects the plane.

(c) double this value of $t$ and substitute it back into $\ell$, this will yield the reflected point.

Explain what you mean by (a)? I'm confused.

Think about it. Two points which are reflected in a plane will lie on a line that is perpendicular to the plane. You can find the parametric equation of this line using your point (10,7,-3) and the perpendicular vector to the plane.

I did all of that.

Here's my entire working out and the answer which I got at the end was incorrect:

Before I check it, what's the answer they give?

Before I check it, what's the answer they give?

any mistakes?

Their answer seems flawed to me, and yours is fine.