5.4 Planes and lines practice 1

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]E: 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?

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 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

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

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]?

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?

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

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”