1.7 Planes in Space

Planes in Space

Introduction

Planes in Space

We will consider what happens when we allow movement in $\R^3$ with the restriction that the movement must be orthogonal to $\vv=\langle 1,2,3 \rangle$.

Find values for $a_0$ and $b_0$ such that $\langle 0,a_0,b_0 \rangle$ is orthogonal to $\langle 1,2,3\rangle$.

Work through this exercise and explain your reasoning step by step.

Write a scalar equation of the plane $p_1$ passing through the point $(0, 2, 4)$ and perpendicular to the vector $\vn=\langle 2, -1, 1\rangle$.

Is the point $(2, 0, 2)$ on the plane $p_1$?

Write a scalar equation of the plane $p_2$ that is parallel to $p_1$ and passes through the point $(3, 0, 4)$.

Give parametric equations for the line $\mathcal{L}$ passing through the point $(2,0,2)$ and perpendicular to the plane $p_3$ described by the equation $x+2y-2z = 7$.

Find the point at which $\mathcal{L}$ intersects $p_3$.

Work through this exercise and explain your reasoning step by step.

Determine the components of the vectors $\overrightarrow{P_0P_1}$ and $\overrightarrow{P_0P_2}$.

Find a normal vector $\vn$ to $p$.

Find a scalar equation of $p$.

Consider a second plane $q$ with scalar equation $-3(x-1) + 4(y+3) + 2(z-5)=0$. Find two different points on $q$ as well as a vector $\vm$ that is normal to $q$.

What is the angle between planes $p$ and $q$?

Find an equation for the plane that contains both of the lines described in .