8.4 Vector and Parametric Equations as Plane
Chapter
Chapter 8
Section
8.4
calcvecnelsonCh8.4lectureHQ 5 Videos
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}

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.

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.

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.

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?

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)
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).
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)
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}

1.37mins
Q7a

Show that A does not lie on \pi. \displaystyle\begin{array}{c}\pi: \vec{r}=(2,0,1)+s(4,2,-1)+t(-1,1,2), s, t \in \mathbf{R}\\A(0,5,-4)\\ \end{array}

Q7b

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?

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.

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}.

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}.

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

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.)

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.

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.