Determine vector and parametric equations of the plane that contains the points A(1, 2, -1), B(2, 1,1), and C(3, 1, 4).

Find the Cartesian equation of the plane that's defined by A(1, 2, -1), B(2, 1,1), and C(3, 1, 4).

a. Determine the vector, parametric, and symmetric equations of the line passing through points A(-3, 2, 8) and B(4, 3, 9).

b. Determine the vector and parametric equations of the plane containing the points A(-2, 2, 8), B(4, 3, 9), and C(-2, -1, 3).

c. Explain why a symmetric equation cannot exist for a plane.

Determine the vector, parametric, and symmetric equations of the line passing through the points A(7, 1, -2) and perpendicular to the plane with equation 2x - 3y + z -1 =0

Determine the Cartesian equation of each of the following planes:

through the point P(0, 1, -2), with normal \vec{n} = (-1, 3, 3)

Determine the Cartesian equation of each of the following planes:

through the points (3, 0, 1) and (0, 1, -1) and perpendicular to the plane with equation x - y - z + 1 = 0

Determine the Cartesian equation of each of the following planes:

through the points (1, 2, 1) and (2, 1, 4), and parallel to the x-axis.

Determine the Cartesian equation of the plane that passes through the origin and contains the lie \vec{r} = (3, 7, 1) + t (2, 2, 3), t\in \mathbb{R}.

Find the vector and parametric equations of the plane that is parallel to the yz-plane and contains the point A(-1, 2, 1).

Determine the Cartesian equation of the lane that contains the line \displaystyle \vec{r} = (2, 3, 2) + t(1, 1, 4), t\in \mathbb{R} , and the point (4, -3, 2).

Determine the Cartesian equation of the plane that contains the following lines:

\displaystyle L_1: \vec{r} = (4, 4, 5) + t(5, -4, 6), t\in \mathbb{R} and

\displaystyle L_2: (4, 4, 5) + s(2, -3, -4), s\in \mathbb{R}

Determine an equation for the line that is perpendicular to the plane 3x - 2y + z = 1 passing through (2, ,3 -3). give your answer in vector, parametric, and symmetric form.

A plane has 3x + 2y -z + 6= 0 as its Cartesian equation. Determine the vector and parametric equations of this plane.

Determine an equation for the line that has the same x- and z-intercepts as the plane with equation 2x + 5y -z + 7 =0. Give your answer in vector, parametric, and symmetric form.

Determine the vector, parametric, and Cartesian forms of the equation of the plane containing the lines L_1: \vec{r_1} = (3, -4, 1) + s(1, -3, -5) and L_2: \vec{r_2} = (7, -1, 0) +t (2, -6, -10) where s, t \in \mathbb{R}

Sketch the plane: 2x + 3y - 6z - 12 = 0

Sketch the plane: 2x + 3y - 12 = 0

Sketch the plane: x -3z - 6 = 0

Sketch the plane: y - 2z - 4 = 0

Sketch the plane: 2x + 3y - 6z = 0

Determine the vector, parametric, and Cartesian equations of each of the following planes.

passing through the points P(1, -2, 5) and Q(3, 1, 2) and parallel to the line with equation L:\vec{r}= 2t\vec{i} + (4t _3)\vec{j} + (t + 1)\vec{k}, t\in \mathbb{R}.

Determine the vector, parametric, and Cartesian equations of each of the following planes.

containing the point A(1, 1, 2) and perpendicular to the line joining the points B(2, 1, -6) and C(-2, 1, 5)

Determine the vector, parametric, and Cartesian equations of each of the following planes.

passing through the points (4, 1, -1) and (5, -2, 4) and parallel to the z-axis.

Determine the vector, parametric, and Cartesian equations of each of the following planes.

passing through the points (1, ,3 -5), (2, 6, 4), and (3, -3, 3)

Show that L_1: \vec{r} = (1, 2, 3) + s(-3, 5, 21)+t(0, 1, 3), s, t \in \mathbb{R}, and L_2: \vec{r} = (1, -1. -6) + u(1, 1, 1) + v(2, 5, 11), u,v, \in \mathbb{R}, are equations for the same plane.

The two lines L_1: \vec{r_1} = (-1, 1, 0) + s(2, 1, -1) s\in \mathbb{R}, and L_2: \vec{r_2} = (2, 1, 2) + t(2, 1, -1), t \in \mathbb{R}, are parallel but do no coincide. The point A(5, ,4 -3) is on L_1. Determine the coordinates of a point B on L_2 such that \vec{AB} is perpendicular to L_2.

Write a brief description of each plane.

2x - 3y = 6

Write a brief description of each plane.

x -3z = 6

Write a brief description of each plane.

2y - z = 6

Which of the following points lies on the line x = 2t, y = 3+ t, z = 1 + t?

A(2, 4, 2), B(-2, 2,1), C(4, 5, 2), D(6, 6, 2)

If the point (a, b, -3) lies on the line, x = 2t, y = 3+ t, z = 1 + t, determine the values of a and b.

Calculate the acute angle that is formed by the intersection of each pair of lines.

L_1: \frac{x -1}{1} = \frac{y-3}{5} and L_2: \frac{x - 2}{2} = \frac{1- y}{3}

Calculate the acute angle that is formed by the intersection of each pair of lines.

y = 4x + 2 and y = -x + 3

Calculate the acute angle that is formed by the intersection of each pair of lines.

L_1: x =-1 + 3t, y = 1+ 4t, z =-2t and L_2: x = -1 + 2s, y = 3s, z = -7 + 2

Calculate the acute angle that is formed by the intersection of each pair of lines.

L_1: (x, y, z) = (4, 7, -1) + t(4, 8, -4) and L_2: (1, 5, 4) + s(-1, 2, 3)

Calculate the acute angle that is formed by the intersection of each pair of planes.

2x + 3y -z + 9 = 0 and x + 2y + 4 = 0

Calculate the acute angle that is formed by the intersection of each pair of planes.

x - y - z - 1 = 0 and 2x + 3y - z + 4 = 0

a. Which of the following lines is parallel to the plane 4x + y - z - 10 = 0?
i. \vec{r} = (3, 0, 2) + t(1, -2, 2) ii. x = -3t, y = -5 + 2t, z = -10t iii. \frac{x - 1}{4} = \frac{y + 6}{-1} =\frac{z}{1}

b. do any of these lines lie in the lane in part a.?

Explain why any plane with a vector equation of the form (x, y, z) = (a, b, c) + s(d, e, f)+ t(a, b, c) will always pass through the origin.