91Ó°ÊÓ

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. [T] \(4 x+5 y+10 z-20=0\)

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. \(3 x+4 y-12=0\)

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. \(3 x-2 y+4 z=0\)

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. \(x+z=0\)

Given point \(P(1,2,3)\) and vector \(\mathbf{n}=\mathbf{i}+\mathbf{j},\) find point \(Q\) on the \(x\) -axis such that \(\overrightarrow{P Q}\) and \(\mathbf{n}\) are orthogonal.

Show there is no plane perpendicular to \(\mathbf{n}=\mathbf{i}+\mathbf{j}\) that passes through points \(P(1,2,3)\) and \(Q(2,3,4)\)

Find parametric equations of the line passing through point \(P(-2,1,3)\) that is perpendicular to the plane of equation \(2 x-3 y+z=7\)

Find symmetric equations of the line passing through point \(P(2,5,4)\) that is perpendicular to the plane of equation \(2 x+3 y-5 z=0\)

show that line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}\) is parallel to plane \(x-2 y+z=6\)

Find the real number \(\alpha\) such that the line of parametric equations \(\quad x=t, y=2-t, z=3+t\) , \(t \in \mathbb{R} \quad\) is parallel to the plane of equation \(a x+5 y+z-10=0\)