8.4 Vector and Parametric Equations as Plane
Chapter
Chapter 8
Section
8.4
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Solutions 18 Videos

State which of the following equations define lines and which define planes. Explain how you made your decision.

a) \vec{r} = (1, 2, 3) + s(1, 1, 0) + t(3, 4, -6), s, t \in \mathbb{R}

b) \vec{r} = (-2, 3, 0) + m(3, 4, 7), m \in \mathbb{R}

c) x = -3 - t, y = 5, z= 4 + t, t \in \mathbb{R}

d) \vec{r} = m(4, -1, 2) + t(4, -1, 5), m, t \in \mathbb{R}

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0.25mins
Q1

A plane has a vector equation \vec{r} = (2, 1, 3) + s(\displaystyle{\frac{1}{3}}, -2, \displaystyle{\frac{3}{4}}) + t(6, -12, 30), s, t \in \mathbb{R}.

a) Express the first direction vector with only integers.

b) Reduce the second direction vector.

c) Write a new equation for the plane using the calculations from part a. and b.

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1.04mins
Q2

A plane has x = 2m, y = -3m + 5n, z = - 1 - 3m - 2n, m, n \in \mathbb{R} as its parametric equations.

a) By inspection, identify the coordinates of a point that is on this plane.

b) What are the direction vectors for this plane?

c) What point corresponds to the parameter values of m = -1 and n = -4?

d) What are the parametric values corresponding to the point A(0, 15, -7)?

e) Using your answer for part d., explain why the point B(0, 15, -8) cannot be on this plane.

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2.48mins
Q3

A plane passes through the points P(-2, ,3, 1), Q(-2, 3, 2), and R(1, 0 , 1).

a) Using \vec{PQ} and \vec{PR} as direction vectors, write a vector equation for this plane.

b) Using \vec{QR} and one other direction vector, write a second vector equation for this plane.

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1.15mins
Q4

Explain why the equation \vec{r} = (-1, 0, -1) + s(2, 3, -4) + t(4, 6, -8), s, t, \in \mathbb{R}, does not represent the equation of a plane. What does this equation represent?

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0.49mins
Q5

Determine vector equations and the corresponding parametric equations of the plane.

  • the plane with direction vectors \vec{a} = (4, 1, 0) and \vec{b} = (3, 4, -1), passing through the point A(-1, 2, 7)
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0.42mins
Q6a

Determine vector equations and the corresponding parametric equations of the plane.

  • the plane passing through the points A(1, 0, 0), B(0, 1, 0), and C(0, 0, 1).
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0.58mins
Q6b

Determine vector equations and the corresponding parametric equations of the plane.

  • the plane passing through points A(1, 1, 0) and B(4, 5, -6), with direction vector \vec{a} = (7, 1, 2)
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0.54mins
Q6c

a) Determine parameters corresponding to the point P(5, 3, 2), where P is a point on the plane with equation

\pi: \vec{r} = (2, 0 , 1) + s(4, 2, -1) + t(-1, 1, 2), s, t, \in \mathbb{R}

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1.37mins
Q7a

A plane has \vec{r} = (-3, 5, 6) + s(-1, 1, 2) + v(2, 1, -3), s, v \in \mathbb{R} as its equation.

a) Give the equations of two intersecting lines that lie on this plane.

b) What point do these two lines have in common?

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0.46mins
Q8

Determine the coordinates of the point where the plane with equation \vec{r} = (4, 1, 6) + s(11, -1, 3) + t(-7, 2, -2), s, t \in \mathbb{R}, crosses the z-axis.

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2.09mins
Q9

Determine the equation of the plane that contains the point P(-1, 2, 1) and the line \vec{r}= (2, 1, 3) + s(4, 1, 5), s\in \mathbb{R}.

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0.58mins
Q10

Determine the equation of the plane that contains the point A(-2, 2, 3) and the line \vec{r} = m(2, -1, 7), m \in\mathbb{R}.

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0.48mins
Q11

a) Determine two pairs of direction vectors that can be used to represent the xy-plane in \mathbb{R}^3

b) Write a vector and parametric equations for the xy-plane in \mathbb{R}^3

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1.22mins
Q12

Show that the following equations represent the same plane:

a) \vec{r} = u(-2, 2, 4) + v(-4, 7, 1), u, v \in \mathbb{R} and

b) \vec{r} = s(-1, 5, -3) + t(-1, -5, 7), s, t \in \mathbb{R}

(Hint: Express each direction vector in the first equation as a linear combination of the direction vectors in the second equation.)

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3.46mins
Q14

The plane with equation \vec{r} = (1, 2, 3) + m(1, 2, 5) + n(1, -1, 3) intersects the y- and z-axes at the points A and B, respectively. Determine the equation of the line that contains these two points.

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3.03mins
Q15

Suppose that the lines L_1 and L_2 are defined by the equations \vec{r} = \vec{OP_o} + s\vec{a} and \vec{r} = \vec{OP_o} + t\vec{b}, respectively, where s, t \in \mathbb{R} and \vec{a} and \vec{b} are non-linear vectors. Prove that the plane defined by the equation \vec{r} = \vec{OP_o} + s\vec{a} + t\vec{b} contains both of these lines.

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0.43mins
Q16
Lectures 5 Videos

Introduction to vector and parametric equations of a plane

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2.38mins
Introduction to vector and parametric equations of a plane

Converting to Parametric Form from Vector Equation of Planes

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1.50mins
Converting to Parametric Form from Vector Equation of Planes

Finding vector equation of a plane given three points

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2.40mins
Finding vector equation of a plane given three points

Finding vector and parametric equaations of a plane given two vectors and a point that lies in the plane ex1

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1.39mins
Finding vector and parametric equaations of a plane given two vectors and a point that lies in the plane ex1

Finding vector and parametric equaations of a plane given two vectors and a point that lies in the plane ex2

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1.13mins
Finding vector and parametric equaations of a plane given two vectors and a point that lies in the plane ex2