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

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

(c) $\mathbf{p}=(3,2,1,4), \mathbf{q}=(2,0,4,1)$

(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: $$\begin{aligned} &x=3 t+2 \\ &y=t+1 \\ &z=-2 t+3 \end{aligned}$$

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

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 $P$.