Question

Let L be the line in R³ that passes through the point (1,−2,4) and is parallel to v=(3,1,−5). Let P be the plane defined by 4x−3y+2z=15. Find the parametric equations of the line, L.

Answer 1

The parametric equations for the line L in R³ that is parallel to v=(3,1,−5) and passes through the point (1,−2,4) are x=1+3t, y=−2+t, and z=4−5t.

Step-by-step explanation:

To find the parametric equations of the line L that passes through the point (1,−2,4) and is parallel to v=(3,1,−5), we use the point-direction form of a line in R³. Since the line is parallel to vector v, the direction vector of the line will be the same as v. The parametric equations of the line can be expressed in terms of a parameter t.

The parametric equations for a line in R³ are given by:

x = x0 + aty = y0 + btz = z0 + ct Where (x0, y0, z0) is a point on the line, and (a, b, c) are the components of the direction vector. In this case, (x0, y0, z0) = (1, −2, 4) and (a, b, c) = (3, 1, −5). Therefore, the parametric equations are:x = 1 + 3ty = −2 + tz = 4 − 5t