The spatial linear direction equation is in the form of (x-x0)/l = (y-y0)/m = (z-z0)/n, and its direction vector is (l, m, n) or inverse (-l, -m, -n).

Such as a straight line

{x+2y-z=7

-2x+y+z=7

(1) First, find an intersection point and take any value of z to solve x and y..

Let z = 0.

From x+2y = 7

-2x+y=7

X =-7/5，y = 2 1/5。

So (-7/5,21/5,0) is a point on a straight line.

(2) Find the direction vector

Because the normal vectors of two known planes are (1, 2,-1), (-2, 1, 1) respectively, the direction vector of the straight line is perpendicular to these two normal vectors.

The direction vector can be obtained by the outer product = (1, 2,-1) × (-2, 1).

=ijk

12- 1

-2 1 1

=3i+j+5k

So the vector in the straight line direction is (3, 1, 5).

Extended data:

The direction of a straight line in space is represented by a non-zero vector parallel to the straight line, which is called the direction vector of the straight line. The position of a straight line in space is completely determined by the space point it passes through and its direction vector.

Given the fixed point P0 (x0, y0, z0) and the non-zero vector v = {l, m, n}, the past point P and the parallel straight line L, V are determined, so the two elements are determined by the straight line point P0 l and V, and V is called the direction vector of L. ..

Because there is no requirement for the length of vectors, the number of direction vectors is infinite for each straight line. Any vector on a straight line is parallel to the direction vector of the straight line.

References:

Baidu Encyclopedia-Direction Vector