Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If I have a horizontal line (a 3d point and 3d vector with zero z component) and another plane (could be an oblique or a horizontal; i have normal vector of the plane); then how do we get the direction (3d) of the 3d line which lie on top of the plane.

For that, I wish to project the above horizontal line on to the given plane.

(I made more clear the original post.)

share|improve this question

1 Answer 1

up vote 0 down vote accepted

I'm not sure what a 2d vector is. I'm assuming you're specifying the line by the span of some vector $v$ translated by the 3d point $p$: $L = p + vt.$

You can specify a plane by two vectors and a point, or by a point and a vector. For the first, call the two vectors $v_1$ and $v_2$, and the point $q$. The plane is $rv_1 + sv_2 + q$.

If $p + vt$ does not intersect the plane, the projection can be written as a translation. If it does intersect the plane, pick $p$ and $q$ so that they coincide with the intersection of the line and the plane. Change coordinates so that $p=q=0$. Now all you do is project the vector $v$ onto $v_1$ and $v_2$. The projection map is

$$tv\mapsto t\langle v,v_1\rangle v_1 + t\langle v,v_2\rangle v_2.$$

If you want to work with a point $q$ and a single vector $w$ which specifies the plane by $\{x\ |\ \langle x,w\rangle = 0\} + q$, again translate coordinates to the intersection point $p=q=0$. Then project onto the span of $w$, and subtract that new line from the old line:

$$ vt\mapsto vt - t\langle v,w\rangle w. $$

share|improve this answer
    
I think that, with regards to the 2D vector, the OP is using the fact that the line is horizontal (i.e. it's really a 3D vector but with the third coordinate zero). –  Lopsy Feb 19 '12 at 22:26
    
@Neal: sorry i didn't get you clearly. i know normal vector of the plane and the horizontal line (a point and direction of line). wouldn't it be taken by getting cross products of some vectors as i cannot get your previous explanation. –  lenin Feb 19 '12 at 23:42
    
Yes, the a normal vector to a plane can be gotten by taking the cross product of two spanning vectors. –  Neal Feb 19 '12 at 23:51
    
@lopsy That makes sense. I think of vectors and 1-forms as "1D", planes and 2-forms as "2D", and so forth, so referring to a vector by the dimension of its ambient space confused me. –  Neal Feb 19 '12 at 23:53
    
@ Neal: No i want to know how can i get my required line (which lies on the plane) direction by getting the cross product of normal vectors? is it possible? if so, please tell me –  lenin Feb 19 '12 at 23:56

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.