?For the following exercises, the equation of a plane is given.a. Find normal vector n

In the following exercises, points P and Q are given. Let L be the line passing through points P and Q. a. Find the vector equation of line L. b. Find parametric equations of line L. c. Find symmetric equations of line L. d. Find parametric equations of the line segment determined by P and Q. P(4, 0, 5), Q(2, 3, 1)

In the following exercises, points P and Q are given. Let L be the line passing through points P and Q. a. Find the vector equation of line L. b. Find parametric equations of line L. c. Find symmetric equations of line L. d. Find parametric equations of the line segment determined by P and Q. P(?1, 0, 5), Q(4, 0, 3)

For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v. a. Find parametric equations of line L. b. Find symmetric equations of line L. c. Find the intersection of the line with the xy-plane. P(1, ?2, 3), v = ? 1, 2, 3 ?

For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v. a. Find parametric equations of line L. b. Find symmetric equations of line L. c. Find the intersection of the line with the xy-plane. P(3, 1, 5), v = ? 1, 1, 1 ?

For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v. a. Find parametric equations of line L. b. Find symmetric equations of line L. c. Find the intersection of the line with the xy-plane. P(3, 1, 5), \(\mathbf{v}=\overrightarrow{Q R}\), where Q(2, 2, 3) and R(3, 2, 3) Text Transcription: mathbf v=overrightarrow Q R

For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v. a. Find parametric equations of line L. b. Find symmetric equations of line L. c. Find the intersection of the line with the xy-plane. P(2, 3, 0), \(\mathbf{v}=\overrightarrow{Q R}\), where Q(0, 4, 5) and R(0, 4, 6) Text Transcription: mathbf v=overrightarrow Q R

For the following exercises, line L is given. a. Find point P that belongs to the line and direction vector v of the line. Express v in component form. b. Find the distance from the origin to line L. x = 1 + t, y = 3 + t, z = 5 + 4t, t ? ?

For the following exercises, line L is given. a. Find point P that belongs to the line and direction vector v of the line. Express v in component form. b. Find the distance from the origin to line L. ?x = y + 1, z = 2

For the following exercises, line L is given. a. Find point P that belongs to the line and direction vector v of the line. Express v in component form. b. Find the distance from the origin to line L. Find the distance between point A(?3, 1, 1) and the line of symmetric equations x = ?y = ?z.

For the following exercises, line L is given. a. Find point P that belongs to the line and direction vector v of the line. Express v in component form. b. Find the distance from the origin to line L. Find the distance between point A(4, 2, 5) and the line of parametric equations x = ?1 ? t, y = ?t, z = 2, t ? ? .

In the following exercises, points P and Q are given. Let L be the line passing through points P and Q. a. Find the vector equation of line L. b. Find parametric equations of line L. c. Find symmetric equations of line L. d. Find parametric equations of the line segment determined by P and Q. P(?3, 5, 9), Q(4, ?7, 2)

In the following exercises, points P and Q are given. Let L be the line passing through points P and Q. a. Find the vector equation of line L. b. Find parametric equations of line L. c. Find symmetric equations of line L. d. Find parametric equations of the line segment determined by P and Q. P(7, ?2, 6), Q(?3, 0, 6)

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. a. Verify whether lines \(L_{1}\) and \(L_{2}\) are parallel. b. If the lines \(L_{1}\) and \(L_{2}\) are parallel, then find the distance between them. \(L_{1}\) : x = 1 + t, y = t, z = 2 + t, \(\quad\) t \(\in \mathbb{R}\), \(\quad L_{2}\) : x ? 3 = y ? 1 = z ? 3 Text Transcription: L_1 L_2 in real number

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. a. Verify whether lines \(L_{1}\) and \(L_{2}\) are parallel. b. If the lines \(L_{1}\) and \(L_{2}\) are parallel, then find the distance between them. \(L_{1}\) x = 2, y = 1, z = t, \(L_{2}\) : x = 1, y = 1, z = 2 ? 3t, \(\quad\) t \(\in \mathbb{R}\) Text Transcription: L_1 L_2 in real number

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. a. Verify whether lines \(L_{1}\) and \(L_{2}\) are parallel. b. If the lines \(L_{1}\) and \(L_{2}\) are parallel, then find the distance between them.. Show that the line passing through points P(3, 1, 0) and Q(1, 4, ?3) is perpendicular to the line with equation x = 3t, y = 3 + 8t, z = ?7 + 6t, \(\quad t \in \mathbb{R}\). Text Transcription: L_1 L_2 in real number

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. a. Verify whether lines \(L_{1}\) and \(L_{2}\) are parallel. b. If the lines \(L_{1}\) and \(L_{2}\) are parallel, then find the distance between them. Are the lines of equations x = ?2 + 2t, y = ?6, z = 2 + 6t and x = ?1 + t, y = 1 + t, z = t, \(\ t \in \mathbb{R}\), perpendicular to each other? Text Transcription: L_1 L_2 in real number

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. a. Verify whether lines \(L_{1}\) and \(L_{2}\) are parallel. b. If the lines \(L_{1}\) and \(L_{2}\) are parallel, then find the distance between them. Find the point of intersection of the lines of equations x = ?2y = 3z and x = ?5 ? t, y = ?1 + t, z = t ? 11, \(\ t \in \mathbb{R}\). Text Transcription: L_1 L_2 in real number

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. a. Verify whether lines \(L_{1}\) and \(L_{2}\) are parallel. b. If the lines \(L_{1}\) and \(L_{2}\) are parallel, then find the distance between them. Find the intersection point of the x-axis with the line of parametric equations x = 10 + t, y = 2 ? 2t, z = ?3 + 3t, \(\ t \in \mathbb{R}\). Text Transcription: L_1 L_2 in real number

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 = y ? 1 = ?z and \(L_{2}: x-2=-y=\frac{z}{2}\) Text Transcription: L_1 L_2 L_2: x-2=-y=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. \(L_{1}\) : x = 2t, y = 0, z = 3, \(\ t \in \mathbb{R}\) and \(L_{2}\) x = 0, y = 8 + s, z = 7 + s, \(\ s \in \mathbb{R}\) Text Transcription: L_1 L_2 t in real number s in real number

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. \(L_{1}: x=-1+2 t, y=1+3 t, z=7 t\), \(t \in \mathbb{R}\) and \(L_{2}: x-1=\frac{2}{3}(y-4)=\frac{2}{7} z-2\) Text Transcription: L_1 L_2 L_1: x=-1+2t, y=1+3t, z=7t t in R L_2: x-1 = 2/3 (y-4) = 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. \(L_{1}: 3 x=y+1=2 z\) and \(L_{2}: x=6+2 t, y=17+6 t, z=9+3 t\), \(t \in \mathbb{R}\) Text Transcription: L_1 L_2 L_1: 3x=y+1=2z L_2: x=6+2t, y=17+6t, z=9+3t t in 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. Text Transcription: x-2=-y=\frac{z}{2}

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. Text Transcription: t in R

For the following exercises, point P and vector n are given. a. Find the scalar equation of the plane that passes through P and has normal vector n. b. Find the general form of the equation of the plane that passes through P and has normal vector n. P(0, 0, 0), n = 3i ? 2j + 4k

For the following exercises, point P and vector n are given. a. Find the scalar equation of the plane that passes through P and has normal vector n. b. Find the general form of the equation of the plane that passes through P and has normal vector n. P(3, 2, 2), n = 2i + 3j ? k

For the following exercises, point P and vector n are given. a. Find the scalar equation of the plane that passes through P and has normal vector n. b. Find the general form of the equation of the plane that passes through P and has normal vector n. P(1, 2, 3), \(\mathbf{n}=\langle 1,2,3\rangle\) Text Transcription: n = langle 1,2,3 rangle

For the following exercises, point P and vector n are given. a. Find the scalar equation of the plane that passes through P and has normal vector n. b. Find the general form of the equation of the plane that passes through P and has normal vector n. P(0, 0, 0), \(\mathbf{n}=\langle-3,2,-1\rangle\) Text Transcription: n = langle -3,2,-1 rangle

For the following exercises, the equation of a plane is given. a. Find normal vector n to the plane. Express n using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. 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 n to the plane. Express n using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. 3x + 4y ? 12 = 0

For the following exercises, the equation of a plane is given. a. Find normal vector n to the plane. Express n using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. 3x ? 2y + 4z = 0

For the following exercises, the equation of a plane is given. a. Find normal vector n to the plane. Express n using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. x + z = 0

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

Show there is no plane perpendicular to n = i + 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.

Show that line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}\) is parallel to plane x ? 2y + z = 6. Text Transcription: x-1 / 2 = y+1 / 3 = z-2 / 4

Find the real number ? 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 ?x + 5y + z ? 10 = 0. Text Transcription: t in R

For the following exercises, points P, Q, and R are given. a. Find the general equation of the plane passing through P, Q, and R. b. Write the vector equation \(\mathbf{n} \cdot \overrightarrow{P S}=0\) 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, and R. P(1, 1, 1), Q(2, 4, 3), and R(?1, ?2, ?1) Text Transcription: n cdot overrightarrow PS = 0

For the following exercises, points P, Q, and R are given. a. Find the general equation of the plane passing through P, Q, and R. b. Write the vector equation \(\mathbf{n} \cdot \overrightarrow{P S}=0\) 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, and R. P(?2, 1, 4), Q(3, 1, 3), and R(?2, 1, 0) Text Transcription: n cdot overrightarrow PS = 0

Consider the planes of equations x + y + z = 1 and x + z = 0. a. Show that the planes intersect. b. Find symmetric equations of the line passing through point P(1, 4, 6) that is parallel to the line of intersection of the planes.

Consider the planes of equations ?y + z ? 2 = 0 and x ? y = 0. a. Show that the planes intersect. b. Find parametric equations of the line passing through point P(?8, 0, 2) that is parallel to the line of intersection of the planes.

Find the scalar equation of the plane that passes through point P(?1, 2, 1) and is perpendicular to the line of intersection of planes x + y ? z ? 2 = 0 and 2x ? y + 3z ? 1 = 0.

Find the general equation of the plane that passes through the origin and is perpendicular to the line of intersection of planes ?x + y + 2 = 0 and z ? 3 = 0.

Determine whether the line of parametric equations x = 1 + 2t, y = ?2t, z = 2 + t, \(t \in \mathbb{R}\) intersects the plane with equation 3x + 4y + 6z ? 7 = 0. If it does intersect, find the point of intersection. Text Transcription: t in R

Determine whether the line of parametric equations x = 5, y = 4 ? t, z = 2t, \(t \in \mathbb{R}\) intersects the plane with equation 2x ? y + z = 5. If it does intersect, find the point of intersection. Text Transcription: t in R

Find the distance from point P(1, 5, ?4) to the plane of equation 3x ? y + 2z ? 6 = 0.

Find the distance from point P(1, ?2, 3) to the plane of equation (x ? 3) + 2(y + 1) ? 4z = 0.

For the following exercises, the equations of two planes are given. a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. [T] x + y + z = 0, 2x ? y + z ? 7 = 0

For the following exercises, the equations of two planes are given. a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. 5x ? 3y + z = 4, x + 4y + 7z = 1

For the following exercises, the equations of two planes are given. a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. x ? 5y ? z = 1, 5x ? 25y ? 5z = ?3

For the following exercises, the equations of two planes are given. a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. [T] x ? 3y + 6z = 4, 5x + y ? z = 4

Show that the lines of equations x = t, y = 1 + t, z = 2 + t, \(t \in \mathbb{R}\) , and \(\frac{x}{2}=\frac{y-1}{3}=z-3\) are skew, and find the distance between them. Text Transcription: t in R x/2 = y-1 / 3 = z - 3

Show that the lines of equations x = ?1 + t, y = ?2 + t, z = 3t, \(t \in \mathbb{R}\) , and x = 5 + s, y = ?8 + 2s, z = 7s, \(s \in \mathbb{R}\) are skew, and find the distance between them. Text Transcription: t in R s in R

Consider point C(?3, 2, 4) and the plane of equation 2x + 4y ? 3z = 8. a. Find the radius of the sphere with center C tangent to the given plane. b. Find point P of tangency.

Consider the plane of equation x ? y ? z ? 8 = 0. a. Find the equation of the sphere with center C at the origin that is tangent to the given plane. b. Find parametric equations of the line passing through the origin and the point of tangency.

Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in the air, curves 3 ft to the right, and falls 5 ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.

[T] John allocates d dollars to consume monthly 302. three goods of prices a, b, and c. In this context, the budget equation is defined as ax + by + cz = d, where \(x \geq 0\), \(y \geq 0\), and \(z \geq 0\) represent the number of items bought from each of the goods. The budget set is given by \(\{(x, y, z) \mid a x+b y+c z \leq d, x \geq 0, y \geq 0, z \geq 0\}\) , and the budget plane is the part of the plane of equation ax + by + cz = d for which \(x \geq 0\), \(y \geq 0\), and \(z \geq 0\). Consider a = $8, b = $5, c = $10, and d = $500. a. Use a CAS to graph the budget set and budget plane. b. For z = 25, find the new budget equation and graph the budget set in the same system of coordinates. Text Transcription: x geq 0 y geq 0 z geq 0 {(x, y, z) mid ax + by + cz leq d, x geq 0, y geq 0, z geq 0}

[T] Consider \(\mathbf{r}(t)=\langle\sin t, \cos t, 2 t\rangle\) the position vector of a particle at time \(t \in[0,3]\), where the components of r are expressed in centimeters and time is measured in seconds. Let \(\overrightarrow{O P}\) be the position vector of the particle after 1 sec. a. Determine the velocity vector v(1) of the particle after 1 sec. b. Find the scalar equation of the plane that is perpendicular to v(1) and passes through point P. This plane is called the normal plane to the path of the particle at point P. c. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point P. Text Transcription: r(t) = langle sin t, cos t, 2t rangle t in [0,3] overrightarrow OP

[T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) A(8, 0, 0), B(8, 18, 0), C(0, 18, 8), and D(0, 0, 8) (see the following figure). a. Find the general form of the equation of the plane that contains the solar panel by using points A, B, and C, and show that its normal vector is equivalent to \(\overrightarrow{A B} \times \overrightarrow{A D}\) . b. Find parametric equations of line \(L_{1}\) that passes through the center of the solar panel and has direction vector \(\mathbf{s}=\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\), which points toward the position of the Sun at a particular time of day. c. Find symmetric equations of line \(L_{2}\) that passes through the center of the solar panel and is perpendicular to it. d. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines \(L_{1}\) and \(L_{2}\) . Text Transcription: overrightarrow AB times overrightarrow AD L_1 L_2 s = 1/sqrt 3 i + 1/sqrt 3 j + 1/sqrt 3 k