Question 1a

Marks: 2

A plane $Π$ contains the point $A (3, 9, - 1)$ and has a normal vector $(\begin{matrix} 4 \\ - 2 \\ 2 \end{matrix}) .$

a)

Find the equation of the plane in its Cartesian form.

Key Concepts

Equation of a Plane in Cartesian Form

Question 1b

Marks: 2

A second point $B$ has coordinates $(- 4, 1, - 3)$ .

b)

Determine whether point B lies on the same plane.

Key Concepts

Equation of a Plane in Cartesian Form

Question 2a

Marks: 3

A plane $Π$ has equation $r = (\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}) + λ (\begin{matrix} - 2 \\ 5 \\ 3 \end{matrix}) + μ (\begin{matrix} 5 \\ 2 \\ 7 \end{matrix}) .$

A line with equation $r = (\begin{matrix} 6 \\ - 2 \\ 1 \end{matrix}) + β (\begin{matrix} 4 \\ 0 \\ 3 \end{matrix})$ intersects $Π$ at a point $Q$ .

a)

Write down the equations of the line and the plane in their parametric forms.

Key Concepts

Equation of a Line in Parametric Form
Equation of a Plane in Vector Form

Question 2b

Marks: 5

b)

Given that the coordinates of

Q

are

(\begin{matrix} 10, & - 2, 4 \end{matrix})

, find the values for

β, λ

and

μ

at the point of intersection.

Key Concepts

Systems of Linear Equations
Intersection of Line & Plane

Question 3a

Marks: 2

Consider the two planes $Π_{1}$ and $Π_{2}$ which can be defined by the equations

$Π_{1} : x + 2 y - z = 5$

$Π_{2} : - 3 x - y + 8 z = 1$

a)

Write down expressions for the normal vectors of each of the two planes.

Key Concepts

Equation of a Plane in Cartesian Form

Question 3b

Marks: 5

b)

Hence find the angle between the two planes. Give your answer in radians.

Key Concepts

Angle Between Two Planes

Question 4a

Marks: 2

The points $A, B$ and $C$ have position vectors $a, b$ and $c$ respectively, relative to the origin $O$ .

The position vectors are given by

$a = 2 i + 3 j - k$

$b = - i + 2 j + 2 k$

$c = i - 4 j + 3 k$

a)

Find the direction vectors

\vec{AB}

and

\vec{AC}

.

Key Concepts

Displacement Vectors
Position Vectors

Question 4b

Marks: 2

Points $A$ , $B$ and $C$ all lie on a single plane.

b)

Use the results from part (a) to write down the vector equation of the plane.

Key Concepts

Equation of a Plane in Vector Form

Question 4c

Marks: 4

c)

Find the Cartesian equation of the plane.

Key Concepts

Equation of a Plane in Cartesian Form

Question 5a

Marks: 2

A plane lies parallel to the line with equation $r = (\begin{matrix} 2 \\ - 2 \\ - 1 \end{matrix}) + β (\begin{matrix} 3 \\ 9 \\ 1 \end{matrix})$ and contains the points $P$ and $X$ with coordinates $(5, 4, 5)$ and $(- 2, 2, 0)$ respectively.

a)

Find the vector

\vec{PX}

.

Key Concepts

Position Vectors
Displacement Vectors

Question 5b

Marks: 2

b)

By appropriate use of the vector product, find the normal to the plane.

Key Concepts

The Vector ('Cross') Product
Equation of a Plane in Vector Form

Question 5c

Marks: 2

c)

Hence find the Cartesian equation of the plane.

Key Concepts

Equation of a Plane in Cartesian Form

Question 6a

Marks: 3

Consider the plane defined by the Cartesian equation $5 x - 3 y - z = 13 .$

a)

Show that the line with equation

r = (\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}) + λ (\begin{matrix} 1 \\ 4 \\ - 7 \end{matrix})

lies in the plane.

Key Concepts

Intersection of Line & Plane

Question 6b

Marks: 3

b)

Show that the line with Cartesian equation

x - 2 = \frac{y - 6}{2} = 2 - z

is parallel to the plane but does not lie in the plane.

Key Concepts

Intersection of Line & Plane
Equation of a Line in Cartesian Form

Question 7a

Marks: 3

Consider the planes $Π_{1}, Π_{2}$ and $Π_{3}$ , which are defined by the equations

$Π_{1} : 3 x - 5 y + z = 27$

$Π_{2} : - 4 x + y + 2 z = - 10$

$Π_{3} : - 2 x - y - z = - 1$

a)

By solving the system of equations represented by the three planes show that the system of equations has a unique solution.

Key Concepts

Systems of Linear Equations
Number of Solutions to a System

Question 7b

Marks: 1

b)

Hence write down the coordinates of any point(s) where all three planes intersect.

Key Concepts

Intersection of Planes
Systems of Linear Equations

Question 8a

Marks: 4

Consider the line $L$ with vector equation $r = (1 - λ) i + (λ - 2) j + (3 + 2 λ) k$ and the plane $Π$ with Cartesian equation $3 x - 2 y + z = 11$ .

a)

Find the angle in radians between the line

L

and the normal to the plane

Π

.

Key Concepts

Angle Between Line & Plane

Question 8b

Marks: 2

b)

Hence find the angle in radians between the line

L

and the plane

Π

.

Key Concepts

Angle Between Line & Plane

Question 9a

Marks: 2

Two planes $Π_{1}$ and $Π_{2}$ are defined by the equations

$Π_{1} : 3 x - 2 y + 4 z = 18$

$Π_{2} : - 2 x + y + 2 z = 7$

a)

Write down expressions for the normal vectors of each of the two planes.

Key Concepts

Equation of a Plane in Cartesian Form

Question 9b

Marks: 2

b)

Find the cross product of the two normal vectors.

Key Concepts

The Vector ('Cross') Product

Question 9c

Marks: 3

c)

Find the coordinates of a point that lies on both planes.

Key Concepts

Systems of Linear Equations
Intersection of Line & Plane

Question 9d

Marks: 2

d)

Hence find a vector equation of the line of intersection of the two planes.

Key Concepts

Intersection of Line & Plane

Question 10a

Marks: 4

A line $L_{1}$ is defined by the Cartesian equation $\frac{x}{3 d + 1} = \frac{y - 3}{4} = 5 - z$ and a plane $Π$ is defined by the Cartesian equation $- x + d y - 4 z = - 29$ , where $d$ is a real constant.

The line $L_{1}$ lies in the plane $Π$ .

a)

Use the fact that the line

L_{1}

lies in the plane

Π

to find the value of the constant $d$ .

Key Concepts

Intersection of Line & Plane
Equation of a Line in Vector Form

Question 10b

Marks: 2

Another line, $L_{2}$ , passes through the origin and is perpendicular to the plane $Π$ .

b)

Write down the equation of line

L_{2}

in vector form.

Key Concepts

Equation of a Line in Vector Form

Question 10c

Marks: 3

c)

By considering the parametric form of the equation for

L_{2}

, or otherwise, determine the point of intersection between line

L_{2}

and the plane

Π

.

Key Concepts

Equation of a Line in Parametric Form
Intersection of Line & Plane

Question 10d

Marks: 2

d)

Hence determine the minimum distance between the plane

Π

and the origin.

Key Concepts

3D Coordinate Geometry
Shortest Distance Between a Point and a Line

DP IB Maths: AA HL

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3.11 Vector Planes

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