$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.2

Projection (Proj) and Perpendicular (Perp)

6 Activities

Wize University Linear Algebra Textbook > Equations of Lines and Planes

Distances

5 Activities

Wize University Linear Algebra Textbook > Equations of Lines and Planes

Lines in $\reals^n$

6 Activities

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0), find the minimum distance of P2\bcb{P_2}P2​ to the line L\bcb{L}L that is parallel to v⃗\bcb{\vec{v}}v that passes through P1\bcb{P_1}P1​. 

Minimum distance =

I don't know

More Projection (Proj) and Perpendicular (Perp) Questions:

Example: Dot Product (2)

Find the scalar and vector projection of b⃗=⟨2,1,3⟩\vec{b}=\langle 2,1,3\rangleb=⟨2,1,3⟩onto a⃗=⟨−1,4,2⟩\vec{a}=\langle -1,4,2\ranglea=⟨−1,4,2⟩.

Find the vector projection of  [3,2][3,2][3,2] onto the vector [3,6]\left[3,6\right][3,6]

133 - FML 3 - 18.1W - e.g. 9

Find the vector-component ı^∣∣d⃗ \ihat_{||_{\vd}} \,^∣∣d​​ of ı^\bcb{\boldsymbol{ \ihat}}^ in the direction of the line given by y=mx\bcb{\boldsymbol{ y=mx}}y=mx. Repeat for ȷ^∣∣d⃗ \jhat_{||{\vd}} \, ^​∣∣d​, the vector-component of ȷ^\bcb{\boldsymbol{ \jhat }}^​ in the direction of the 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>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0)
Find the point on the line L\bcb{L}L closest to the point P2\bcb{P_2}P2​.

133 - FML 3 - 18.1W e.g. 58.2

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0)
Find the minimum distance of P2\bcb{P_2}P2​ to the line L\bcb{L}L found above. 

133 - FML 3 - 18.1W e.g. 48

 Find a unit vector in the direction of the projection of u⃗=<102,112,−56>\bcb{\vec{u} = \left< 102, 112, -56 \right>}u=⟨102,112,−56⟩ onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}v=⟨0,3,4⟩.

133 - FML 3 - 18.1W e.g. 43

 Given u⃗=<8,2,2>\bcb{\vec{u} = \left< 8, 2, 2 \right>}u=⟨8,2,2⟩ and v⃗=<2,−1,−1>\bcb{\vec{v} = \left< 2, -1, -1 \right>}v=⟨2,−1,−1⟩, find the vector projection of u⃗\bcb{\vec{u}}u onto v⃗\bcb{\vec{v}}v. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41

Let P=(−3,−3,−4)\bcb{P = (-3, -3, -4)}P=(−3,−3,−4), Q=(−5,−6,−3)\bcb{Q = (-5,-6,-3)}Q=(−5,−6,−3), R=(−6,−4,−6)\bcb{R = (-6,-4,-6)}R=(−6,−4,−6) and S=(−2,−3,−4)\bcb{S = (-2, -3, -4)}S=(−2,−3,−4). Let l1\bcb{l_1}l1​ be the line passing through P\bcb{P}P and Q\bcb{Q}Q, and let l2\bcb{l_2}l2​ be the line passing through R\bcb{R}R and S\bcb{S}S. Find the distance between R\bcb{R}R and l1\bcb{l_1}l1​. 

133 - FML 3 - 18.1W e.g. 36

For two lines having directional vectors v1\mathbf{v}_1v1​ and v2\mathbf{v}_2v2​, respectively, the shortest distance between them can be found by creating a vector that connects the two lines (by subtracting a point on one line from a point on the other), i.e. (r2−r1)\left(\mathbf{r}_2 - \mathbf{r}_1\right)(r2​−r1​), and projecting it onto the unit normal vector connecting the two lines. The unit normal is easily found by crossing their respective directionals and dividing by the magnitude of the resulting vector, as illustrated in Fig. 5. i.e.
s=∣projn^(r2−r1)∣=∣(r2−r1)⋅v1×v2∣v1×v2∣∣
 	s = \left| \text{proj}_{\mathbf{\hat{n}}} \left(\mathbf{r}_2 - \mathbf{r}_1\right)\right| = \left| \left(\mathbf{r}_2 - \mathbf{r}_1\right) \cdot \frac{\mathbf{v}_1 \times \mathbf{v}_2}{\left|\mathbf{v}_1 \times \mathbf{v}_2\right|}\right|
 s=∣projn^​(r2​−r1​)∣=​(r2​−r1​)⋅∣v1​×v2​∣v1​×v2​​​
Find the equations of the line passing through the point (2,3,−4)\bcb{(2, 3, -4)}(2,3,−4) parallel to the vector i−4k\bcb{\mathbf{i} - 4\mathbf{k}}i−4k ; express it  in vector-parametric, scalar-parametric and standard form. 
 

$\tkct{cut from 19.4F}$ 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)}P1​=(0,0,1) and  P2=(1,−2,3)\bcb{P_2 = (1,-2,3)}P2​=(1,−2,3) and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}v=⟨2,−3,1⟩, find the point Po\bcb{P_o}Po​ on the plane Π: 2x−3y+z=1\bcb{\Pi: \ 2x - 3y + z = 1}Π: 2x−3y+z=1 that is closest to the point P2\bcb{P_2}P2​.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 48

 Find a unit vector in the direction of the projection of u⃗=<102,112,−56>\bcb{\vec{u} = \left< 102, 112, -56 \right>}u=⟨102,112,−56⟩ onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}v=⟨0,3,4⟩.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 43

 Given u⃗=<8,2,2>\bcb{\vec{u} = \left< 8, 2, 2 \right>}u=⟨8,2,2⟩ and v⃗=<2,−1,−1>\bcb{\vec{v} = \left< 2, -1, -1 \right>}v=⟨2,−1,−1⟩, find the vector projection of u⃗\bcb{\vec{u}}u onto v⃗\bcb{\vec{v}}v. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.2

Given the points P1=(0,0,1)\bcb{P_1 = (0,0,1)}P1​=(0,0,1) and  P2=(1,−2,3)\bcb{P_2 = (1,-2,3)}P2​=(1,−2,3) and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}v=⟨2,−3,1⟩
Find the minimum distance between the point P2\bcb{P_2}P2​ and the plane Π: 2x−3y+z=1\bcb{\Pi:\ 2x-3y + z = 1}Π: 2x−3y+z=1. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.3

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0), find the point on 
the line L\bcb{L}L (which is parallel to v⃗\bcb{\vec{v}}v that passes through P1\bcb{P_1}P1​) that is closest to the point P2\bcb{P_2}P2​.

Prove formula for distance between point and plane

Let a,b,c,d,p,q,r∈Ra,b,c,d,p,q,r\in\mathbb{R}a,b,c,d,p,q,r∈R be constants.  Prove that the formula for the distance between the point Q⃗=(p,q,r)\vec Q=(p,q,r)Q​=(p,q,r) and the plane ax+by+cz=dax+by+cz=dax+by+cz=d is ∣d−ap−bq−cr∣a2+b2+c2\frac{|d-ap-bq-cr|}{\sqrt{a^2+b^2+c^2}}a2+b2+c2​∣d−ap−bq−cr∣​ .

Find the closest distance of point (2,2,2) to the line L by deriving and solving system of linear equations.
L:{𝑥1+x2+𝑥3=2𝑥1−𝑥2−2x3=5L:\begin{cases}
𝑥_1+ x_2 + 𝑥_3 = 2\\
𝑥_1 − 𝑥_2 − 2x_3 = 5

\end{cases}L:{x1​+x2​+x3​=2x1​−x2​−2x3​=5​

Let L1L_1L1​ be a line through the point (1,3,1)(1,3,1)(1,3,1) with direction vector
[112]T\begin{bmatrix}1&1&2\end{bmatrix}^T[1​1​2​]Tand let L2L_2L2​ be a line through the point (2,1,2)(2,1,2)(2,1,2) with direction vector [113]T\begin{bmatrix}1&1&3\end{bmatrix}^T[1​1​3​]T . Find the shortest distance from L1L_1L1​ to L2L_2L2​.

133 - FML 3 - 18.1W e.g. 41

Let P=(−3,−3,−4)\bcb{P = (-3, -3, -4)}P=(−3,−3,−4), Q=(−5,−6,−3)\bcb{Q = (-5,-6,-3)}Q=(−5,−6,−3), R=(−6,−4,−6)\bcb{R = (-6,-4,-6)}R=(−6,−4,−6) and S=(−2,−3,−4)\bcb{S = (-2, -3, -4)}S=(−2,−3,−4). Let l1\bcb{l_1}l1​ be the line passing through P\bcb{P}P and Q\bcb{Q}Q, and let l2\bcb{l_2}l2​ be the line passing through R\bcb{R}R and S\bcb{S}S. Find the distance between R\bcb{R}R and l1\bcb{l_1}l1​. Find the distance between l1\bcb{l_1}l1​ and l2\bcb{l_2}l2​. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.8

 Given the planes
π1 : 4x−3y+z=2\bcb{\pi_1 \, : \, 4x - 3y + z = 2}π1​:4x−3y+z=2 and π2 : x+2y+2z=0\bcb{\pi_2 \, : \, x + 2y + 2z = 0}π2​: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.}L1​:⎩⎨⎧​xyz​===​3t+2−2t+1t​, L2​:⎩⎨⎧​xyz​===​−2t+1t+1t+3​, L3​:⎩⎨⎧​xyz​===​t+3−t2t+1​,

Shortest Distance Between Two Lines, Parallel Edition

Find the shortest distance between the following two lines in R3\mathbb{R}^3R3:
L1={x=1+2ty=3−tz=2+3t,L2={1−4t2+2t−6tL_1 = \begin{cases}
x = 1+2t \\
y = 3-t \\
z = 2+3t
\end{cases}, L_2 = \begin{cases}
1-4t \\
2+2t \\
-6t
\end{cases}L1​=⎩⎨⎧​x=1+2ty=3−tz=2+3t​,L2​=⎩⎨⎧​1−4t2+2t−6t​

Shortest Distance Between Point and Plane

Find the shortest distance between the point (1,2,3)(1,2,3)(1,2,3) and the plane 2x−3y+z=42x-3y+z=42x−3y+z=4.

Shortest Distance Between Point and Line

Find the shortest distance between the point (1,2,3)(1,2,3)(1,2,3) and the line L⃗(t)=<2t+3,t+4,−t−2>\vec{L}(t)=\left<2t+3,t+4,-t-2\right>L(t)=⟨2t+3,t+4,−t−2⟩.

Shortest Distance Between Two Planes

Find the shortest distance between the following two planes in R3\mathbb{R}^3R3:
P1:2x−3y+z=1P2:−6x+9y−3z=5P_1: 2x - 3y + z = 1\\
P_2: -6x + 9y - 3z = 5P1​:2x−3y+z=1P2​:−6x+9y−3z=5

Shortest Distance Between Two Lines, Parallel Edition

Find the shortest distance between the following two lines in R3\mathbb{R}^3R3:
L1={x=1+2ty=3−tz=2+3t,L2={1−4t2+2t−6tL_1 = \begin{cases}
x = 1+2t \\
y = 3-t \\
z = 2+3t
\end{cases}, L_2 = \begin{cases}
1-4t \\
2+2t \\
-6t
\end{cases}L1​=⎩⎨⎧​x=1+2ty=3−tz=2+3t​,L2​=⎩⎨⎧​1−4t2+2t−6t​

Shortest Distance Between Two Lines

Find the shortest distance between the following two lines in R3\mathbb{R}^3R3:
L1={x=1+2ty=3−tz=2+3t,L2={−t2+t−3tL_1 = \begin{cases}
x = 1+2t \\
y = 3-t \\
z = 2+3t
\end{cases}, L_2 = \begin{cases}
-t \\
2+t \\
-3t
\end{cases}L1​=⎩⎨⎧​x=1+2ty=3−tz=2+3t​,L2​=⎩⎨⎧​−t2+t−3t​

Shortest Distance Between Plane and Line

Find the shortest distance between the following plane and line in R3\mathbb{R}^3R3:
P:2x−3y+z=1,L:{x=2t+3y=t+4z=−t−2P: 2x-3y+z=1, L: \begin{cases}
x = 2t+3 \\
y = t + 4 \\
z = -t - 2
\end{cases}P:2x−3y+z=1,L:⎩⎨⎧​x=2t+3y=t+4z=−t−2​

Shortest Distance

Find the shortest distance from the point (3, −1, 2)\left(3,\ -1,\ 2\right)(3, −1, 2) to the plane 2x−y+z=102x-y+z=102x−y+z=10.

More Lines in $\reals^n$ 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}y−z=42x+y+3z=2​

a) Write the parametric equation of line 
L:{x+2y=0x−z=5L: \begin{cases}
x+2y=0\\
x-z=5
\end{cases}L:{x+2y=0x−z=5​
b) Find intersection point of the line L with the plane P:3y−3z=2.P:3y-3z=2.P:3y−3z=2.

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)(1,2,−3) 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)l1​: (−5, 0, 1)+t(1,0,−1) and l2: (3,−1,6)+t(0,2,1)l_2:\ \left(3,-1,6\right)+t\left(0,2,1\right)l2​: (3,−1,6)+t(0,2,1).

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)(x, y, z)=(1−t)(6,−1,2)+t(9,5,−7).

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)(1,2,−3) 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)(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)(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=-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)}(2,3,−4) parallel to the vector i−4k\bcb{\mathbf{i} - 4\mathbf{k}}i−4k. 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>}w=⟨1,−2,1⟩ that contains the point P=(−1,1,2)\bcb{P = (-1, 1, 2)}P=(−1,1,2). 

Lines in $\reals^n$

Practice: Lines in Rn\colorOne{\reals^n}Rn
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>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0)
Find the point on the line L\bcb{L}L closest to the point P2\bcb{P_2}P2​.

133 - FML 3 - 18.1W e.g. 58.2

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0)
Find the minimum distance of P2\bcb{P_2}P2​ to the line L\bcb{L}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)}P1​=(1,0,5) and P2=(3,1,−2)\bcb{P_2 = (3, 1, -2)}P2​=(3,1,−2), which of the following points are on the line connecting P1\bcb{P_1}P1​ and P2\bcb{P_2}P2​?

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41

Let P=(−3,−3,−4)\bcb{P = (-3, -3, -4)}P=(−3,−3,−4), Q=(−5,−6,−3)\bcb{Q = (-5,-6,-3)}Q=(−5,−6,−3), R=(−6,−4,−6)\bcb{R = (-6,-4,-6)}R=(−6,−4,−6) and S=(−2,−3,−4)\bcb{S = (-2, -3, -4)}S=(−2,−3,−4). Let l1\bcb{l_1}l1​ be the line passing through P\bcb{P}P and Q\bcb{Q}Q, and let l2\bcb{l_2}l2​ be the line passing through R\bcb{R}R and S\bcb{S}S. Find the distance between R\bcb{R}R and l1\bcb{l_1}l1​. 

$\tkct{cut from 19.4F}$ 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>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0), find the scalar-parametric equation of the line L\bcb{L}L  parallel to v⃗\bcb{\vec{v}}v that passes through P1\bcb{P_1}P1​.

133 - FML 3 - 18.1W - e.g. 8

If a line is given by y=mx\bcb{\boldsymbol{ y = mx}}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)A(0,−3) and B(−3,6)B(-3,6)B(−3,6).

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.2

Related Topics

Projection (Proj) and Perpendicular (Perp)

Distances

Lines in Rn\reals^nRn

More Projection (Proj) and Perpendicular (Perp) Questions:

Example: Dot Product (2)

133 - FML 3 - 18.1W - e.g. 9

133 - FML 3 - 18.1W e.g. 58.3

133 - FML 3 - 18.1W e.g. 58.2

133 - FML 3 - 18.1W e.g. 48

133 - FML 3 - 18.1W e.g. 43

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41

133 - FML 3 - 18.1W e.g. 36

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.3

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 48

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 43

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.2

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.3

More Distances Questions:

Prove formula for distance between point and plane

133 - FML 3 - 18.1W e.g. 41

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.8

Shortest Distance Between Two Lines, Parallel Edition

Shortest Distance Between Point and Plane

Shortest Distance Between Point and Line

Shortest Distance Between Two Planes

Shortest Distance Between Two Lines, Parallel Edition

Shortest Distance Between Two Lines

Shortest Distance Between Plane and Line

Shortest Distance

More Lines in Rn\reals^nRn Questions:

Find vec form line thru pt ortho to 2 vec form lines

Find vec form line thru pt ortho to 2 vec form lines

133 - FML 3 - 18.1W e.g. 33; 222/262_mid_#2_19.1W_eg5

133 - FML 3 - 18.1W e.g. 50

Lines in $\reals^n$

133 - FML 3 - 18.1W e.g. 58.3

133 - FML 3 - 18.1W e.g. 58.2

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 56

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.1

133 - FML 3 - 18.1W - e.g. 8

Equation of Lines

Lines in $\reals^n$

More Lines in $\reals^n$ Questions: