Answer to Question #290597 in Analytic Geometry for abed

Question

We consider the point M(1, 0, 2) and the lines d1 :    x = t + 1, y = −t + 2, z = t − 1 and d2 : x = y = z. (a) Find the director vectors of the lines d1 and d2. (b) Find the angle between the lines d1 and d2. (c) Write the equation of the plane passing through the point M that is perpendicular to the line d2

Answer 1

a) direction vector are the coefficient of t

Direction vector of d_1:

_<1,-1,1>

Direction vector of d₂:

x=t, y=t, z=t is:

<1,1,1>

b) <1,-1,1> . <1,1,1> =|<1,-1,1>||<1,1,1>|cos$\theta$

$1=\sqrt{3}×\sqrt{3}\>cos\theta$

$\theta=70.5°$

c) Normal to the plane is <1,1,1>

The equation of the plane is of the form

x+y+z=d

It passes through (1,0,2)

$\therefore$ 1+0+2=d

d=3

The equation of the plane is

x+y+z=3