직선과 평면의 방정식

For the following exercises, lines ${L}_{1}$ and ${L}_{2}$ are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

${L}_{1}:x=-1+2t,y=1+3t,z=7t,$ $t\in \mathbb{R}$ and ${L}_{2}:x-1=\frac{2}{3}(y-4)=\frac{2}{7}z-2$

For the following exercises, lines ${L}_{1}$ and ${L}_{2}$ are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

${L}_{1}:3x=y+1=2z$ and ${L}_{2}:x=6+2t,y=17+6t,z=9+3t,$ $t\in \mathbb{R}$

Consider line $L$ of symmetric equations $x-2=-y=\frac{z}{2}$ and point $A(1,1,1).$

a. Find parametric equations for a line parallel to $L$ that passes through point $A.$ b. Find symmetric equations of a line skew to $L$ and that passes through point $A.$ c. Find symmetric equations of a line that intersects $L$ and passes through point $A.$

Consider line $L$ of parametric equations $x=t,y=2t,z=3,$ $t\in \mathbb{R}.$

a. Find parametric equations for a line parallel to $L$ that passes through the origin. b. Find parametric equations of a line skew to $L$ that passes through the origin. c. Find symmetric equations of a line that intersects $L$ and passes through the origin.

For the following exercises, point $P$ and vector $\text{n}$ are given.

a. Find the scalar equation of the plane that passes through $P$ and has normal vector $\text{n}.$ b. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\text{n}.$

$P(0,0,0),$ $\text{n}=3\text{i}-2\text{j}+4\text{k}$

For the following exercises, point $P$ and vector $\text{n}$ are given.

a. Find the scalar equation of the plane that passes through $P$ and has normal vector $\text{n}.$ b. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\text{n}.$

$P(3,2,2),$ $\text{n}=2\text{i}+3\text{j}-\text{k}$

For the following exercises, point $P$ and vector $\text{n}$ are given.

a. Find the scalar equation of the plane that passes through $P$ and has normal vector $\text{n}.$ b. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\text{n}.$

$P(1,2,3),$ $\text{n}=\langle 1,2,3\rangle$

For the following exercises, point $P$ and vector $\text{n}$ are given.

a. Find the scalar equation of the plane that passes through $P$ and has normal vector $\text{n}.$ b. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\text{n}.$

$P(0,0,0),$ $\text{n}=\langle -3,2,-1\rangle$

For the following exercises, the equation of a plane is given.

a. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors. b. Find the intersections of the plane with the coordinate axes. c. Sketch the plane.

[T] $4x+5y+10z-20=0$

For the following exercises, the equation of a plane is given.

a. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors. b. Find the intersections of the plane with the coordinate axes. c. Sketch the plane.

$3x+4y-12=0$

For the following exercises, the equation of a plane is given.

a. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors. b. Find the intersections of the plane with the coordinate axes. c. Sketch the plane.

$3x-2y+4z=0$

For the following exercises, the equation of a plane is given.

a. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors. b. Find the intersections of the plane with the coordinate axes. c. Sketch the plane.

$x+z=0$

Given point $P(1,2,3)$ and vector $\text{n}=\text{i}+\text{j},$ find point $Q$ on the x-axis such that $\vec{PQ}$ and $\text{n}$ are orthogonal.

Show there is no plane perpendicular to $\text{n}=\text{i}+\text{j}$ that passes through points $P(1,2,3)$ and $Q(2,3,4).$

Find parametric equations of the line passing through point $P(-2,1,3)$ that is perpendicular to the plane of equation $2x-3y+z=7.$

Find symmetric equations of the line passing through point $P(2,5,4)$ that is perpendicular to the plane of equation $2x+3y-5z=0.$

For the following exercises, the equation of a plane is given.

a. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors. b. Find the intersections of the plane with the coordinate axes. c. Sketch the plane.

Show that line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ is parallel to plane $x-2y+z=6.$

Find the real number $\alpha$ such that the line of parametric equations $x=t,y=2-t,z=3+t,$ $t\in \mathbb{R}$ is parallel to the plane of equation $\alpha x+5y+z-10=0.$

For the following exercises, points $P,Q,\,\text{and}\,R$ are given.

a. Find the general equation of the plane passing through $P,Q,\,\text{and}\,R.$ b. Write the vector equation $\text{n}\cdot \vec{PS}=0$ of the plane at a., where $S(x,y,z)$ is an arbitrary point of the plane. c. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through $P,Q,\,\text{and}\,R.$

$P(1,1,1),Q(2,4,3),$ and $R(-1,-2,-1)$

For the following exercises, points $P,Q,\,\text{and}\,R$ are given.

a. Find the general equation of the plane passing through $P,Q,\,\text{and}\,R.$ b. Write the vector equation $\text{n}\cdot \vec{PS}=0$ of the plane at a., where $S(x,y,z)$ is an arbitrary point of the plane. c. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through $P,Q,\,\text{and}\,R.$

$P(-2,1,4),Q(3,1,3),$ and $R(-2,1,0)$