Answer to Question #214395 in Analytic Geometry for prince

Question

(1.1) Let U and V be the planes given by: U : λx + 5y − 2λz − 3 = 0,

V : −λx + y + 2z + 1 = 0.

Determine for which value(s) of λ the planes U and V are:

(a) orthogonal,

(b) Parallel.

(1.2) Find an equation for the plane that passes through the origin (0, 0, 0) and is parallel to the (3) plane −x + 3y − 2z = 6.

(1.3) Find the distance between the point (−1, −2, 0) and the plane 3x − y + 4z = −2.

Answer 1

1.

a)

$-\lambda^2+5-4\lambda=0$

$\lambda_1=1, \lambda_2=4$

b)

The planes are parallel when

$\lambda/-\lambda=5=-2\lambda/2$

So. such $\lambda$ does not exist.

2.

−x + 3y − 2z =0

3.

$d=\frac{|-3+2+2|}{\sqrt{3^2+1+4^2}}=1/\sqrt{26}$