Planes in Space

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

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

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

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 $x+2y-2z = 7$.

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

Let $P_0 = (1,2,-1)$, $P_1 = (1, 0 ,-1)$, and $P_2 = (0,1,3)$ and let $p$ be the plane containing $P_0$, $P_1$, and $P_2$.

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

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

3. Find a scalar equation of $p$.

4. 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 $q$.

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

This exercise explores key relationships between a pair of planes. Consider the following two planes: one with scalar equation $4x - 5y + z = -2$, and the other which passes through the points $(1,1,1)$, $(0,1,-1)$, and $(4, 2, -1)$.

1. Find a vector normal to the first plane.

2. Find a scalar equation for the second plane.

3. Find the angle between the planes, where the angle between them is defined by the angle between their respective normal vectors.

4. …

1. Find the plane that goes through the points $(1,2,3)$, $(4,5,6)$, and $(7,8,9)$ or explain why there is not one plane that satisfies this condition.

2. Find all planes in $\mathbb{R}^3$ whose line of intersection with the $xz$-plane is given by $\pi x +e z=\sqrt{5}$.

3. Give two vectors that are in the plane given by $3x+4y=6z-20$ and are orthogonal to each other.

4. Give the equations of two planes that make a $90^\circ$ angle with $3x+4y=6z-20$ and have an angle of $90^\circ$ between them as well.

5. Give the p…

In this problem, we explore how we can use what we know about vectors and projections to find the distance from a point to a plane.

Let $p$ be the plane with equation $z=-4x+3y+4$, and let $Q = (4,-1,8)$.

1. Show that $Q$ does not lie in the plane $p$.

2. Find a normal vector $\vn$ to the plane $p$.

3. Find the coordinates of a point $P$ in $p$.

4. Find the components of $\overrightarrow{PQ}$. Draw a picture to illustrate the objects found so far.

5. Explain why $|\comp_{\vn} \overrightarrow{PQ}|$ gives the di…