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$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.7

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Wize University Linear Algebra Textbook > Equations of Lines and Planes

Planes in R3\reals^3

6 Activities

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  1. Grab a piece of paper and try this problem yourself.
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Given the planes
π1 : 4x−3y+z=2\bcb{\pi_1 \, : \, 4x - 3y + z = 2} and π2 : x+2y+2z=0\bcb{\pi_2 \, : \, x + 2y + 2z = 0}, and the lines
L1 : {x=3t+2y=−2t+1z=t, L2 : {x=−2t+1y=t+1z=t+3, L3 : {x=t+3y=−tz=2t+1\bcb{L_1 \, : \, \left\{ \begin{matrix} x & = & 3t + 2 \\ y & = & -2t + 1 \\ z & = & t \end{matrix} \right., ~ L_2 \, : \, \left\{ \begin{matrix} x & = & -2t + 1 \\ y & = & t + 1 \\ z & = & t + 3 \end{matrix} \right., ~ L_3 \, : \, \left\{ \begin{matrix} x & = & t + 3 \\ y & = & -t \\ z & = & 2t +1 \end{matrix} \right.},
and the point P1=(1,−1,2)\bcb{P_1 = (1,-1,2)}.

Find the intersection of the planes π1\bcb{\pi_1} and π2\bcb{\pi_2}.
More Planes in R3\reals^3 Questions:
Write down the equation of a plane defined by this parametric equation: P: (-2,0,5)+s(3,1,2)+t(-2,2,0)
Consider the two lines, L_1:(1,0,0)+s(1,1,1) and L_2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Write down equation of Line L: (1,3,5)+t(-4,1,3) as the intersection of two planes.
Find a point whose distance to the plane 3x+4y−10z=303x+4y-10z=30 is equal to 10.
Consider two planes with equations 3x+2y−z=8 and ax−6y+bz=8.3x+2y-z=8\ and\ ax-6y+bz=8.
a) For what values of a and b are two planes parallel?
b) For what values of a and b is the angle between two planes 60 degrees?
If two planes 2x+ay−2z=2y+62x+ay-2z=2y+6 and x=y+z+3x=y+z+3 are parallel, at which point does the first plane intersect with x-axis.
Consider two lines: L1:(1,0,0)+s(1,1,1) and L2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Find gen form plane thru pt parallel to 2 vecs
Given that the plane PP passes through the point (1,3,−1)\left(1,3,-1\right) and is parallel to both vectors
𝑢⃗=(0,1,1)𝑢⃗ = (0,1,1) and 𝑣⃗=(3,2,1),\vec𝑣 = (3,2,1),find the standard form equation of the plane.
(type your answer in the form ax+by+cz=d, with no spaces)
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.1
Given the points P1=(0,0,1)\bcb{P_1 = (0,0,1)} and P2=(1,−2,3)\bcb{P_2 = (1,-2,3)} and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}
Find the equation of the plane Π\bcb{\Pi} perpendicular to v⃗\bcb{\vec{v}}, containing the point P1\bcb{P_1}.
Planes in $\reals^3$
Practice: Normal Vector
Which of the following vectors is normal to the plane 3x+z=73x+z=7?
A plane is parallel to the vector v=⟨1,1,0⟩v = \langle1, 1, 0\rangle. If it passes through points (1,2,−1)(1,2,-1) and (2,1,0)(2,1,0) then find the equation of the plane.
Which one of the following is a vector that is orthogonal to the line that passes through the points P(1,−1,0)P\left(1,-1,0\right) and Q(5,0,−3)Q\left(5,0,-3\right)?
Which of the following is a vector orthogonal/ perpendicular to the plane 2x−z=32x-z=3?
133 - FML 3 - 18.1W e.g. 30
Find the equation of a plane passing through the point (2,0,1)\bcb{(2, 0, 1)}, perpendicular to the straight line passing through the points (1,1,0)\bcb{(1, 1, 0)} and (4,−1,−2)\bcb{(4, -1, -2)}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.4
Given the planes
π1 : 4x−3y+z=2\bcb{\pi_1 \, : \, 4x - 3y + z = 2} and π2 : x+2y+2z=0\bcb{\pi_2 \, : \, x + 2y + 2z = 0}, and the lines
L1 : {x=3t+2y=−2t+1z=t, L2 : {x=−2t+1y=t+1z=t+3, L3 : {x=t+3y=−tz=2t+1\bcb{L_1 \, : \, \left\{ \begin{matrix} x & = & 3t + 2 \\ y & = & -2t + 1 \\ z & = & t \end{matrix} \right., ~ L_2 \, : \, \left\{ \begin{matrix} x & = & -2t + 1 \\ y & = & t + 1 \\ z & = & t + 3 \end{matrix} \right., ~ L_3 \, : \, \left\{ \begin{matrix} x & = & t + 3 \\ y & = & -t \\ z & = & 2t +1 \end{matrix} \right.},
The following linear equations represent planes in R3R^3. Describe the intersection of these planes (if any).
2x+y−z=3x−y−z=33x+z=6\begin{array}{c} 2x+y-z=3\\ x-y-z=3\\ 3x+z=6 \end{array}
Planes in $\reals^3$
Practice: Planes in R3\colorOne{\reals^3} (Tricky)
Consider the points A(−1,3,2)A(-1,3,2), B(−3,5,0)B(-3,5,0), and C(xC,yC,zC)C(x_C,y_C,z_C), and suppose we know the vector AC→=⟨3,−4,−5⟩\overrightarrow{AC}=\lang 3,-4,-5\rang.
Find the equation of the plane containing point CC, and perpendicular to the line passing through points AA and BB.
133 - FML 3 - 18.1W e.g. 31
Find a point on, and a vector normal to, the plane 2x+y+3z=6\bcb{2x + y + 3z = 6}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 57
Find the equation of a plane that contains the three points P1=(1,−2,3)\bcb{P_1 = (1,-2, 3)}, P2=(0,−1,4)\bcb{P_2 = (0, -1, 4)} and P3=(4,3,−2)\bcb{P_3 = (4,3,-2)}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 50
Find the equation of a plane perpendicular to w⃗=<1,−2,1>\bcb{\vec{w} = \left< 1, -2, 1\right>} that contains the point P=(−1,1,2)\bcb{P = (-1, 1, 2)}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.3
Given the points P1=(0,0,1)\bcb{P_1 = (0,0,1)} and P2=(1,−2,3)\bcb{P_2 = (1,-2,3)} and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}, find the point Po\bcb{P_o} on the plane Π: 2x−3y+z=1\bcb{\Pi: \ 2x - 3y + z = 1} that is closest to the point P2\bcb{P_2}.
133 - FML 3 - 18.1W e.g. 31
Find a point on, and a vector normal to, the plane 2x+y+3z=6\bcb{2x + y + 3z = 6}.
Planes in R^3
Which one of the following is the standard form equation of the plane in R3\reals^3 which contains the lines (x,y,z)=(2,0,−1)+t(2,−5,−1)(x,y,z)=(2,0,-1)+t(2,-5,-1) and (x,y,z)=(3,1,2)+t(3,−5,1)(x,y,z)=(3,1,2)+t(3,-5,1)