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a) Write the parametric equation of line L: x+2y=0 x-z=5 b) Find intersectio…

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Wize University Linear Algebra Textbook > Equations of Lines and Planes

Intersection of a Line and a Plane

5 Activities

Wize University Linear Algebra Textbook > Equations of Lines and Planes

Lines in Rn\reals^n

6 Activities

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a) Write the parametric equation of line

L:{x+2y=0x−z=5L: \begin{cases} x+2y=0\\ x-z=5 \end{cases}

b) Find intersection point of the line L with the plane P:3y−3z=2.P:3y-3z=2.
More Intersection of a Line and a Plane Questions:
Find the intersection of the line L and the plane P given below
𝐿:(1,1,0)+𝑡(1,0,−2)𝐿:(1,1,0)+𝑡(1,0,−2)
𝑃:𝑥+𝑦+𝑧=3𝑃:𝑥+𝑦+𝑧=3
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.3
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. 60.1
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 and planes
Practice: Intersection of lines and planes
Find the point of intersection of the line L: [1,2,0]+𝑡[−1,1,1]L:\ \left[1,2,0\right]+𝑡\left[−1,1,1\right] and the plane Π: 𝑥+𝑦−3𝑧=0.\Pi:\ 𝑥+𝑦−3𝑧=0.
Intersection of Lines and Planes
Practice: Intersection of Lines & Planes
Find the point of intersection of the line and the plane:
l: x⃗=⟨1,2,0⟩+t⟨2,1,1⟩l: \ \vec x = \lang 1,2,0\rang + t\lang 2,1,1\rang
Intersection of Lines and Planes
Practice: Intersection of Lines & Planes
Find the point(s) of intersection between the following line and plane:
l:{x=1+ty=1−tz=3+2tl:\begin{cases} x=1+t\\y=1-t\\z=3+2t \end{cases}
More Lines in Rn\reals^n Questions:
Consider u=(1,2,3) and v=(2,-1,4)
a. Find a parametric equation of the line passing through the point (1,3,1) and
perpendicular to both u and v.
1) Find a vector of length 1 that is parallel to the line given by the equations:
𝑦−𝑧=42𝑥+𝑦+3𝑧=2\begin{array}{}𝑦−𝑧=4\\2𝑥+𝑦+3𝑧=2 \end{array}
Find vec form line thru pt ortho to 2 vec form lines
Find the equation of the line that passes through the point (1,2,−3)\left(1,2,-3\right) and is orthogonal to both lines
l1: (−5, 0, 1)+t(1,0,−1)l_1:\ \left(-5,\ 0,\ 1\right)+t\left(1,0,-1\right) and l2: (3,−1,6)+t(0,2,1)l_2:\ \left(3,-1,6\right)+t\left(0,2,1\right).
Practice Question: Two Point Form Equation
Write a vector that is parallel to the line (x, y, z)=(1−t)(6,−1,2)+t(9,5,−7)\left(x,\ y,\ z\right)=\left(1-t\right)\left(6,-1,2\right)+t\left(9,5,-7\right).
Find vec form line thru pt ortho to 2 vec form lines
Find the equation of the line that passes through the point (1,2,−3)\left(1,2,-3\right) and is orthogonal to the lines with equations:
(x,y,z)=(−5,0,1)+t(1,0,−1)(x,y,z)= (-5,0,1)+t(1,0,-1) and (x,y,z)=(3,−1,6)+t(0,2,1)(x,y,z)=(3,-1,6)+t(0,2,1)
Which of the following vectors is a normal vector of a line that is parallel to 2x+y=−32x+y=-3?
133 - FML 3 - 18.1W e.g. 33; 222/262_mid_#2_19.1W_eg5
Find the equations of the line passing through the point (2,3,−4)\bcb{(2, 3, -4)} parallel to the vector i−4k\bcb{\mathbf{i} - 4\mathbf{k}}. Express them in vector-parametric, scalar-parametric and standard form.
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)}.
Lines in $\reals^n$
Practice: Lines in Rn\colorOne{\reals^n}
Various pieces of information can be given to find the equation of a line.
133 - FML 3 - 18.1W e.g. 58.3
Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}
Find the point on the line L\bcb{L} closest to the point P2\bcb{P_2}.
133 - FML 3 - 18.1W e.g. 58.2
Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}
Find the minimum distance of P2\bcb{P_2} to the line L\bcb{L} found above.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 56
Given the points P1=(1,0,5)\bcb{P_1 = (1, 0, 5)} and P2=(3,1,−2)\bcb{P_2 = (3, 1, -2)}, which of the following points are on the line connecting P1\bcb{P_1} and P2\bcb{P_2}?
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41
Let P=(−3,−3,−4)\bcb{P = (-3, -3, -4)}, Q=(−5,−6,−3)\bcb{Q = (-5,-6,-3)}, R=(−6,−4,−6)\bcb{R = (-6,-4,-6)} and S=(−2,−3,−4)\bcb{S = (-2, -3, -4)}. Let l1\bcb{l_1} be the line passing through P\bcb{P} and Q\bcb{Q}, and let l2\bcb{l_2} be the line passing through R\bcb{R} and S\bcb{S}. Find the distance between R\bcb{R} and l1\bcb{l_1}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.1
Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}, find the scalar-parametric equation of the line L\bcb{L} parallel to v⃗\bcb{\vec{v}} that passes through P1\bcb{P_1}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.2
Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}, find the minimum distance of P2\bcb{P_2} to the line L\bcb{L} that is parallel to v⃗\bcb{\vec{v}} that passes through P1\bcb{P_1}.
133 - FML 3 - 18.1W - e.g. 8
If a line is given by y=mx\bcb{\boldsymbol{ y = mx}}, find a unit vector pointing in the direction of the line (i.e. find the directional vector).
Equation of Lines
Which of of the following is an equation of the line that passes through A(0,−3)A(0,-3) and B(−3,6)B(-3,6).