1.4.E: Lines, Planes, and Hyperplanes (Exercises)

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Exercise \(\PageIndex{1}\)

Find vector and parametric equations for the line in \(\mathbb{R}^2\)through \(\mathbf{p}=(2,3)\) in the direction of \(\mathbf{v}=(1,-2)\).

Answer

Vector equation: \(\mathbf{y}=t(1,-2)+(2,3)=(t+2,-2 t+3)\)

Parametric equations:

\[\begin{aligned}
&x=t+2 \\
&y=-2 t+3
\end{aligned}\]

Exercise \(\PageIndex{2}\)

Find vector and parametric equations for the line in \(\mathbb{R}^4\)through \(\mathbf{p}=(1,-1,2,3)\) in the direction of \(\mathbf{v}=(-2,3,-4,1)\).

Exercise \(\PageIndex{3}\)

Find vector and parametric equations for the lines passing through the following pairs of points.

(a) \(\mathbf{p}=(-1,-3), \mathbf{q}=(4,2)\)

(b) \(\mathbf{p}=(2,1,3), \mathbf{q}=(-1,2,1)\)

(d) \(\mathbf{p}=(4,-3,2), \mathbf{q}=(1,-2,4)\)

Answer

(a) Vector equation: \(\mathbf{y}=t(5,5)+(-1,-3)=(5 t-1,5 t-3)\)

Parametric equations: \[ \begin{aligned}
&x=5 t-1 \\
&y=5 t-3
\end{aligned} \]

(b) Vector equation: \(\mathbf{y}=t(3,1,-2)+(2,1,3)=(3 t+2, t+1,-2 t+3)\)

Parametric equations: \[ \begin{aligned}
&x=3 t+2 \\
&y=t+1 \\
&z=-2 t+3
\end{aligned} \]

Parametric equations: \[ \begin{aligned}
w &=t+3 \\
x &=2 t+2 \\
y &=-3 t+1 \\
z &=3 t+4
\end{aligned} \]

(d) Vector equation: \(\mathbf{y}=t(3,-1,-2)+(4,-3,2)=(3 t+4,-t-3,-2 t+2)\)

Parametric equations: \[ \begin{aligned}
&x=3 t+4 \\
&y=-t-3 \\
&z=-2 t+2
\end{aligned} \]

Exercise \(\PageIndex{4}\)

Find the distance from the point \(\mathbf{q}=(1,3)\) to the line with vector equation \( \mathbf{y} = t(2,1) + (3,1) \).

Exercise \(\PageIndex{5}\)

Find the distance from the point \(\mathbf{q}=(1,3,-2)\) to the line with vector equation \( \mathbf{y} = t(2,-1,4)+(1,-2,-1) \).

Answer: \(\frac{1085}{7}\)

Exercise \(\PageIndex{6}\)

Find the distance from the point \(\mathbf{r}=(-1,2,-3)\) to the line through the points \( \mathbf{p} = (1,0,1)\) and \(\mathbf{q}=(0,2,-1)\).

Exercise \(\PageIndex{7}\)

Find the distance from the point \(\mathbf{r}=(-1,-2,2,4)\) to the line through the points \(\mathbf{p}=(2,1,1,2)\) and \(\mathbf{q}=(1,2,-4,3)\).

Answer: \(\frac{4697}{14}\)

Exercise \(\PageIndex{8}\)

Find vector and parametric equations for the plane in \(\mathbb{R}^3\)which contains the points \(\mathbf{p}=(1,3,-1)\), \(\mathbf{q}=(-2,1,1)\), and \(\mathbf{r}=(2,-3,2)\).

Exercise \(\PageIndex{9}\)

Find vector and parametric equations for the plane in \(\mathbb{R}^4\)which contains the points \(\mathbf{p}=(2,-3,4,-1)\), \(\mathbf{q}=(-1,3,2,-4)\), and \(\mathbf{r}=(2,-1,2,1)\).

Answer

Vector equation: \(\mathbf{y}=t(-3,6,-2,-3)+s(0,2,-2,2)+(2,3,4,-1)\)

Parametric equations: \[ \begin{aligned}
&w=-3 t+2 \\
&x=6 t+2 s+3 \\
&y=-2 t-2 s+4 \\
&z=-3 t+2 s-1
\end{aligned} \]

Exercise \(\PageIndex{10}\)

Let \(P\) be the plane in \(\mathbb{R}^3\)with vector equation \(\mathbf{y}=t(1,2,1)+s(-2,1,3)+(1,0,1) \). Find the distance from the point \(\mathbf{q}=(1,3,1)\) to \(P\).

Exercise \(\PageIndex{11}\)

Let \(P\) be the plane in \(\mathbb{R}^4\)with vector equation \( \mathbf{y}=t(1,-2,1,4)+s(2,1,2,3)+(1,0,1,0) \). Find the distance from the point \(\mathbf{q}=(1,3,1,3)\) to \(P\).

Answer: 3

Exercise \(\PageIndex{12}\)

Find a normal vector and a normal equation for the line in \(\mathbb{R}^2\)with vector equation \(\mathbf{y}=t(1,2)+(1,-1) \).

Exercise \(\PageIndex{13}\)

Find a normal vector and a normal equation for the line in \(\mathbb{R}^2\)with vector equation \( \mathbf{y}=t(0,1)+(2,0) \).

Answer

\(\mathbf{n}=(1,0)\)

Normal equation: \((1,0) \cdot(x-2, y)=0 \), or \( x=2\)

Exercise \(\PageIndex{14}\)

Find a normal vector and a normal equation for the plane in \(\mathbb{R}^3\)with vector equation \(\mathbf{y}=t(1,2,1)+s(3,1,-1)+(1,-1,1) \).

Exercise \(\PageIndex{15}\)

Find a normal vector and a normal equation for the line in \(\mathbb{R}^2\)which passes through the points \(\mathbf{p}=(3,2)\) and \(\mathbf{q}=(-1,3)\).

Answer

\(\mathbf{n}=(1,4)\)

Normal equation: \((1,4) \cdot(x-3, y-2)=0, \text { or } x+4 y=11\)

Exercise \(\PageIndex{16}\)

Find a normal vector and a normal equation for the plane in \(\mathbb{R}^3\)which passes through the points \(\mathbf{p}=(1,2,-1)\), \(\mathbf{q}=(-1,3,1)\), and \(\mathbf{r}=(2,-2,2)\).

Answer

\(\mathbf{n}=(11,8,7)\)

Normal equation: \((11,8,7) \cdot(x-1, y-2, z+1)=0, \text { or } 11 x+8 y+7 z=20\)

Exercise \(\PageIndex{17}\)

Find the distance from the point \(\mathbf{q}=(3,2)\) in \(\mathbb{R}^2\)to the line with equation \( x+2 y-3=0 \).

Answer: \(\frac{4}{\sqrt{5}}\)

Exercise \(\PageIndex{18}\)

Find the distance from the point \(\mathbf{q}=(1,2,-1)\) in \(\mathbb{R}^3\)to the plane with equation \( x+2 y-3 x=4 \).

Exercise \(\PageIndex{19}\)

Find the distance from the point \(\mathbf{q}=(3,2,1,1)\) in \(\mathbb{R}^4\)to the hyperplane with equation \( 3 x+y-2 z+3 w=15 \).

Answer: \(\frac{3}{\sqrt{23}}\)

Exercise \(\PageIndex{20}\)

Find the angle between the lines in \(\mathbb{R}^2\)with equations \(3 x+y=4 \) and \( x-y=5 \).

Exercise \(\PageIndex{21}\)

Find the angle between the planes in \(\mathbb{R}^3\)with equations \( 3 x-y+2 z=5 \) and \( x-2 y+z=4\).

Answer: 0.7017 radians

Exercise \(\PageIndex{22}\)

Find the angle between the hyperplanes in \(\mathbb{R}^4\)with equations \( w+x+y-z=3 \) and \( 2 w-x+2 y+z=6 \text { . } \)

Exercise \(\PageIndex{23}\)

Find an equation for a plane in \(\mathbb{R}^3\)orthogonal to the plane with equation \( x+2 y-3 z=4\) and passing through the point \(\mathbf{p}=(1,-1,2)\).

Answer: \(2 x-y=3\) is the equation of one such plane.

Exercise \(\PageIndex{24}\)

Find an equation for the plane in \(\mathbb{R}^3\)which is parallel to the plane \(x-y+2 z=6 \) and passes through the point \(\mathbf{p}=(2,1,2)\).

Exercise \(\PageIndex{25}\)

Show that if \( \mathbf{x}\), \( \mathbf{y}\), and \( \mathbf{z}\) are vectors in \(\mathbb{R}^n\)with \(\mathbf{x} \perp \mathbf{y}\) and \(\mathbf{x} \perp \mathbf{z}\), then \(\mathbf{x} \perp(a \mathbf{y}+b \mathbf{z})\) for any scalars \(a\) and \(b\).

Exercise \(\PageIndex{26}\)

Find parametric equations for the line of intersection of the planes in \(\mathbb{R}^3\)with equations \( x+2 y-6 z=4 \) and \( 2 x-y+z=2 \).

Exercise \(\PageIndex{27}\)

Find parametric equations for the plane of intersection of the hyperplanes in \(\mathbb{R}^4\)with equations \( w-x+y+z=3\) and \( 2 w+4 x-y+2 z=8 \).

Answer: \(y=2 t-\frac{2}{3}, z=-s-t+\frac{11}{3}\)

Exercise \(\PageIndex{28}\)

Let \(L\) be the line in \(\mathbb{R}^3\)with vector equation \( \mathbf{y}=t(1,2,-1)+(3,2,1) \) and let \(P\) be the plane in \(\mathbb{R}^3\)with equation \(x+2 y-3 z=8 \). Find the point where \(L\) intersects \(P\).

Exercise \(\PageIndex{29}\)

Let \(P\) be the plane in \(\mathbb{R}^n\)with vector equation \(\mathbf{y}=t \mathbf{v}+s \mathbf{w}+\mathbf{p} \). Let \(\mathbf{c}\) be the projection of \(\mathbf{w}\) onto \(\mathbf{v}\),

\[ \mathbf{a}=\frac{1}{\|v\|} \mathbf{v}, \nonumber \]

and

\[ \mathbf{b}=\frac{1}{\|\mathbf{w}-\mathbf{c}\|}(\mathbf{w}-\mathbf{c}). \nonumber \]

Show that \( \mathbf{y}=t \mathbf{a}+s \mathbf{b}+\mathbf{p} \) is also a vector equation for \(P\).