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Consider two lines: L1:(1,0,0)+s(1,1,1) and L2:(-1,2,3)+t(4,0,c). a) For what v…

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Wize University Linear Algebra Textbook > Equations of Lines and Planes

Planes in R3\reals^3

6 Activities

Wize University Linear Algebra Textbook > Equations of Lines and Planes

Intersection of Two Lines

6 Activities

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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.
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.
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. 60.7
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.},
$\tkct{cut from 19.4F}$ 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)
More Intersection of Two Lines Questions:
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
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.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.6
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.},
Practice: Intersection of lines
Practice: Intersection of lines
Which of the following statements is/are true about these lines?
L1: [1,1,2]+𝑡[−1,2,3]L_1:\ \left[1,1,2\right]+𝑡\left[−1,2,3\right]
Which of the following statements is true about the lines
L1:x=4+ty=−1+tz=−3−2tL_1:\begin{array}{l}x=4+t\\y=-1+t\\z=-3-2t\end{array}and L2:x=1+ry=2−rz=3−2rL_2:\begin{array}{l}x=1+r\\y=2-r\\z=3-2r\end{array}
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
Find the value of ccif the point (a,b,c)\left(a,b,c\right)is a point of intersection between the lines
L1:x=1+ty=t+2z=3−2tL_1:\begin{array}{l}x=1+t\\y=t+2\\z=3-2t\end{array}and L2:x=−1+2ry=1+rz=4−rL_2:\begin{array}{l}x=-1+2r\\y=1+r\\z=4-r\end{array}
Line Intersections
Consider the lines:
l1:(x1,x2,x3,x4)=(0,−3,−1,2)+s(1,0,2,−1)l_1:(x_1,x_2,x_3,x_4)=(0,-3,-1,2)+s(1,0,2,-1)
l2:(x1,x2,x3,x4)=(6,−4,2,0)+t(−4,1,1,0)l_2:(x_1,x_2,x_3,x_4)=(6,-4,2,0)+t(-4,1,1,0)