5.4 Planes and lines

If points [math]A(6,0,6), B(2,2,0), C(2,0,8)[/math] and [math]D(8,k,6)[/math] are all on plane [math]E[/math], what is the value of [math]k[/math]?

Find the equation of plane [math]E[/math] passing through point [math](6,2,0)[/math] and parallel to plane [math]x-3y+4=0[/math].

Consider plane [math]x-2y-2z=24[/math] and sphere [math]\Gamma[/math] with center [math](9,0,-3)[/math] and radius [math]2[/math]. Which of the following best represents the intersection of [math]\Gamma[/math] and the plane?

(Triple intercepts form) If [math]E[/math] is a plane that passes through [math](a,0,0), (0,b,0)[/math], and [math](0,0,c)[/math] where [math]a, b[/math], and [math]c[/math] are nonzero real numbers, what is the equation of [math]E[/math]?

Find the equations of the following planes.

(a) A plane containing point [math](1,0,4)[/math] and line [math]-x+1=y-2=0.5z[/math].

(b) A plane containing lines [math]\begin{cases}x=2+t\\y=-2t\\z=6+3t\end{cases}, t\in\mathbb{R}[/math] and [math]\begin{cases}x=6-t\\y=4+2t\\z=5-3t\end{cases}, t\in\mathbb{R}[/math].

If [math]L[/math] is a line that passes through points [math](2,4,7), (3,0,4)[/math], what is the equation of [math]L[/math]?
Note: Each box contains a whole number

If [math]L[/math] is a line that passes through point [math](8,1,6)[/math] and is parallel to vector [math]-\textbf{i}+3\textbf{k}[/math], what is the equation of [math]L[/math]?
Note: Each box contains a whole number

The intersection of planes [math]x+3y-z=7[/math] and [math]2x-y+z=2[/math] is a line. What is the equation of the line?
Note: Each box contains a whole number

(Calculator) Find the acute angle between planes [math]E_1: 2x+y-2z=5[/math] and [math]E_2: 4y-5z=10[/math].

Find the acute angle between line [math]x=\dfrac{y-1}{-1}=\dfrac{3z}{2}[/math] and plane [math]x+2y-z=8[/math].

(Calculator) Consider point [math]P(9,2,1)[/math] and plane [math]E:3x+2y-6z=4[/math], find [math]Q\in E[/math] such that [math]PQ[/math] is minimum.

(Calculator) Let [math]L_1:\displaystyle{\frac{x-7}{4}=\frac{2-y}{4}=-(z+6)}[/math] and [math]L_2:\displaystyle{\frac{x+5}{3}=\frac{y-2}{-4}=\frac{z-1}{-2}}[/math] be two skewed lines. If [math]P\in L_1[/math] and [math]Q\in L_2[/math] are points on the common perpendicular line, find the coordinates of [math]P[/math] and [math]Q[/math].

Consider three lines [math]L_1: \begin{cases}x=2-t\\y=3+t\\z=4+2t\end{cases}[/math], [math]L_2: \begin{cases}x=3+2t\\y=17+3t\\z=-4-6t\end{cases}[/math] and [math]L_3: \begin{cases}x=t\\y=1-6t\\z=4\end{cases}[/math].

(a) Find the intersection of [math]L_1[/math] and [math]L_2[/math].

(b) Find the equation of plane [math]E[/math] containing both [math]L_1[/math] and [math]L_2[/math].

(c) [math]L_3[/math] and [math]E[/math] are
(A) parallel
(B) perpendicular
(C) neither parallel nor perpendicular

(d) If the distance between [math]L_3[/math] and [math]E[/math] is [math]\dfrac{a}{\sqrt{b}}[/math] where [math]a[/math] and [math]b[/math] are integers and [math]b[/math] is a prime number, what is the value of [math]a+b[/math]?

(Calculator) Consider line [math]L: \begin{cases}x=2+t\\y=3+2t\\z=5-t\end{cases}, t\in\mathbb{R}[/math] and point [math]P(-8,4,7)[/math].

(a) Find the coordinates of [math]Q\in L[/math] such that [math]\overline{PQ}\perp L[/math].

(b) If the distance between [math]P[/math] and [math]L[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is a rational number, what is the value of [math]a[/math]?

Which of the following conditions cannot determine a unique plane?

Consider planes [math]E_1, E_2[/math], and [math]E_3[/math]. Suppose [math]E_1[/math] is parallel to [math]E_2[/math] but not parallel to [math]E_3[/math]. Which of the following statement(s) is/are true?

Consider pyramid [math]A-BCDE[/math] with vertices [math]B(2,2,0), C(-2,2,0), D(-2,-2,0)[/math], and [math]E(2,-2,0)[/math]. Four triangles [math]ABC, ACD, ADE[/math] and [math]AEB[/math] are all equilateral triangles.

(a) If the coordinates of [math]A[/math] are [math](a, b, \sqrt{c})[/math] where [math]a, b[/math], and [math]c[/math] are integers, what is the value of [math]a+b+c[/math]?

(b) If cosine of the angle between triangles [math]ABC[/math] and [math]ACD[/math] is [math]\dfrac{1}{\sqrt{p}}[/math] where [math]p[/math] is a positive integer, what is the value of [math]p[/math]?

Find the equation of the plane passing through [math](0,0,3), (0,-4,0)[/math], and [math](5,0,0)[/math].

What is the direction vector of the line fromed by the intersection of planes [math]x+y+z=3[/math] and [math]5x-3y+2z=4[/math]?
Note: Each box contains a whole number

Consider line [math]L: \dfrac{x}{2}=\dfrac{y}{-3}=\dfrac{z-1}{2}[/math] and points [math]P(0,0,1)[/math] and [math]Q(2,2,5)[/math]. [math]P[/math] is on [math]L[/math].

(a) Find [math]\text{proj}_{\vec{L}}\overrightarrow{PQ}[/math] where [math]\overrightarrow{L}[/math] is the direction vector of [math]L[/math].

(b) Let [math]L_2[/math] be a line passing through [math]Q[/math] and perpendicular to [math]L[/math]. If [math]L[/math] and [math]L_2[/math] interesect, use (a) to find the equation of [math]L_2[/math].

A fly moves along vector [math]\overrightarrow{u}=3\textbf{i}+2\textbf{j}+4\textbf{k}[/math] from the origin. A butterfly moves along vector [math]\overrightarrow{v}=2\textbf{i}+\textbf{j}+3\textbf{k}[/math] from [math](1,1,1)[/math]. Will they crash into each other? Write “Yes” or “No”

(Calculator) Consider plane [math]E: x-2y+2z=3[/math] and line [math]L: x-5=-y+3=\dfrac{z-9}{2}[/math].

(a) Find the acute angle between [math]L[/math] and [math]E[/math].

(b) Let [math]Q[/math] be the intersection of [math]E[/math] and [math]L[/math]. Find the coordinates of [math]Q[/math].

(c) Consider points [math]P(5,3,9)[/math] and [math]P’\in E[/math]. If [math]\overrightarrow{P’P}\perp\overrightarrow{P’Q}[/math]. Find [math]\|\overrightarrow{P’Q}\|[/math].