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The following linear equations represent planes in R^3. Describe the intersecti…

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Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Geometric Interpretation of Linear Systems

3 Activities

Wize University Linear Algebra Textbook > Equations of Lines and Planes

Planes in R3\reals^3

6 Activities

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}
More Geometric Interpretation of Linear Systems Questions:
What is the reduced row echelon form of an augmented matrix with three equations and three unknowns,
a) If all solutions lie on the line[x y z]=(3,1,0)+s(1,2,3)[x\ y\ z] =(3,1,0)+s(1,2,3)
b) If all solutions lie on the plane x+y+2z=1x+y+2z=1
For what value of p does the following system of equations have no solution? What is the geometric interpretation of these equations?
x+3y=1x+3y=1
px−6y=−4px-6y=-4
Practice Question 6: Solving Systems of Linear Equations
Practice Question: Solving Systems of Linear Equations
The following equations represent three planes in ℝ3. Describe the regions of intersection.
a) x1−x2+2x3=0x_1-x_2+2x_3=0
Solving Linear Systems
Practice: Geometric Interpretation of a Linear System
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
Geometric Interpretation of Linear Systems
Example: Geometric Interpretation of a Linear System
Solve the system of linear equations and interpret the solution geometrically:
2x+2y=6−x−y=−3\begin{array}{rl} 2x+2y&=6\\ -x-y&=-3\\ \end{array}
Homogeneous equations
Find the solution(s) of Ax=0, if the solution of Ax=b is given as follows:
xp=[1030]+s[−1100]+t[00−23]x_p=\begin{bmatrix} 1\\ 0\\ 3\\ 0 \end{bmatrix}+s\begin{bmatrix} -1\\ 1\\ 0\\ 0 \end{bmatrix}+t\begin{bmatrix} 0\\ 0\\ -2\\ 3 \end{bmatrix}
Interpretation of Linear Systems
Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl} 2x_1+3x_3+x_4&=&1\\ x_2-x_4&=&0\\ x_2&=&-3 \end{array}
Homogeneous equations
Find the solution(s) of Ax=0, if the solution of Ax=b is given as follows:
xp=[1030]+s[−1100]+t[00−23]x_p=\begin{bmatrix} 1\\ 0\\ 3\\ 0 \end{bmatrix}+s\begin{bmatrix} -1\\ 1\\ 0\\ 0 \end{bmatrix}+t\begin{bmatrix} 0\\ 0\\ -2\\ 3 \end{bmatrix}
Practice: Solutions to a System of Equations (Step 3)
Practice: Solutions to a System of Equations (Step 3)
Find the solution(s) to the systems of linear equations whose augmented matrices are given:
a.) [1−1030012]\left[ \begin{array}{ccc|c} 1&-1&0&3\\ 0&0&1&2\\ \end{array}\right]
System of Equations
e.g. True or False: if a linear system has more unknowns than equations, then it must have infinitely many solutions. Justify your answer with a proof or counter-example.
System of Equations
True or False: if a linear system has more unknowns than equations, then it must have infinitely many solutions. Justify your answer with a proof or counter-example.
Linear combinations
If u⃗=[5−23]\vec{u}=\begin{bmatrix}5\\-2\\3\end{bmatrix}is a linear combination of v⃗=[−2k3]\vec{v}=\begin{bmatrix}-2\\k\\3\end{bmatrix}and w⃗=[32−1]\vec{w}=\begin{bmatrix}3\\2\\-1\end{bmatrix}, what is the value of kk?
Solving Linear Systems
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
If the unknowns in this system are x1, x2, x3, x4, x5x_1,\ x_2,\ x_3,\ x_4,\ x_5, find the solution(s) to the linear system.
Solving Linear Systems
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
If the unknowns in this system are x1, x2, x3, x4, x5x_1,\ x_2,\ x_3,\ x_4,\ x_5, find the solution(s) to the linear system.
Basics of Systems of Linear Equations
Determine which vectors are solutions to the following SLE [select all that apply]:
x1 − 5x2 +x3=2−x1 +5x2 −3x3= −6\begin{alignedat}{8} x_1\ &-&\ 5x_2\ &+& x_3 &=& 2\\ -x_1\ &+& 5x_2\ &-& 3x_3 &=&\ -6 \end{alignedat}
Interpretation of Linear Systems
Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl} 2x_1+3x_3+x_4&=&1\\ x_2-x_4&=&0\\ x_2&=&-3 \end{array}
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. 60.7
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.},
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)