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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 ex2

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1.13mins