LemonmathSelf-study sheet

Grade 12 · Lesson 7.6

Revision: coordinates in space

Name: Date: Time spent: min

Goals

Calculate vector components, lengths, scalar products and the angle between two vectors quickly.
Write equations of planes, lines and spheres and read their features from an equation.
Use the angle and distance formulas in multiple-choice, true/false and short-answer questions.
Set up a coordinate system for a real-life situation (a room, an aircraft, a radar station).

Exercises

Exercise 1 Multiple choice

The plane through $M(1, -2, 3)$ with normal vector $\vec n=(2, -1, 1)$ has equation

A. $2x-y+z-7=0$ . B. $2x-y+z+7=0$ . C. $x-2y+3z-7=0$ . D. $2x-y+z-3=0$ .

Exercise 2 Multiple choice

The distance from $M(1, 2, 3)$ to the plane $(P)\colon 2x-2y+z+3=0$ is

A. $4$ . B. $\dfrac49$ . C. $\dfrac43$ . D. $\dfrac23$ .

Exercise 3 Multiple choice

The sphere $(S)\colon x^2+y^2+z^2-2x+4y-6z-2=0$ has centre $I$ and radius $R$ given by

A. $I(1, -2, 3)$ , $R=4$ . B. $I(-1, 2, -3)$ , $R=4$ . C. $I(1, -2, 3)$ , $R=2\sqrt3$ . D. $I(2, -4, 6)$ , $R=4$ .

Exercise 4 True/false

Consider the line $d\colon x=1+t$ , $y=2-t$ , $z=2t$ and the plane $(P)\colon x+y+z-3=0$ .

a) $d$ has direction vector $\vec u=(1, -1, 2)$ .

b) The point $A(2, 1, 2)$ lies on $d$ .

c) $d$ is parallel to $(P)$ .

d) $d$ meets $(P)$ at the point $(1, 2, 0)$ .

Exercise 5 Short answer

Find the angle between the lines $d_1\colon x=t$ , $y=1+t$ , $z=2$ and $d_2\colon x=1$ , $y=2+s$ , $z=-s$ (in degrees).

Exercise 6 Short answer

Let $A(1, 0, 1)$ , $B(2, 1, 0)$ , $C(0, 2, 3)$ . Find the distance from the origin $O$ to the plane $(ABC)$ (to 2 decimal places).

Exercise 7 Short answer

A room is a cuboid $6$ m long, $4$ m wide and $3$ m high. Choose axes $Oxyz$ with the origin at a corner of the floor and the axes along three edges, in metres. A straight wire is stretched from the point $A(0, 0, 3)$ in a corner of the ceiling to the point $B(6, 4, 0)$ in the opposite corner of the floor. Find the length of the wire (in m, to 2 decimal places).

Exercise 8 Short answer

A drone starts at the origin $O$ and flies in a straight line; after $t$ seconds it is at $M(2t, 2t, t)$ (in metres). The no-fly zone is the region of space where $x+2y+2z\ge30$ . After how many seconds does the drone start to enter the no-fly zone?

Exercise 9 Written answer

Let $A(3, 1, -2)$ and the plane $(P)\colon x+2y-2z=0$ . Write the equations of the line $d$ through $A$ perpendicular to $(P)$ , find the orthogonal projection $H$ of $A$ on $(P)$ , and find $d(A, (P))$ .

Exercise 10 Short answer (challenge)

The sphere $(S)$ has centre $I(1, 2, -1)$ and touches the plane $(P)\colon 2x-y+2z+11=0$ at the point $H(a, b, c)$ . Find $a+b+c$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. In

Oxyz

, let

A(1, 2, 3)

and

B(3, 0, 4)

. The length of

AB

9

.B.

3

.C.

2\sqrt2

.D.

5

Question 2. A normal vector of the plane

(P)\colon 2x-3z+1=0

(2, -3, 1)

.B.

(2, 0, -3)

.C.

(2, -3, 0)

.D.

(0, 2, -3)

Question 3. The line through

A(1, -1, 2)

with direction vector

\vec u=(2, 1, -1)

has parametric equations

x=1+2t

y=-1+t

z=2-t

.B.

x=2+t

y=1-t

z=-1+2t

.C.

x=1+2t

y=1+t

z=2-t

.D.

x=-1+2t

y=1+t

z=-2-t

Question 4. The angle between the planes

(P)\colon x-z=0

and

(Q)\colon y-z=0

30^\circ

.B.

45^\circ

.C.

60^\circ

.D.

90^\circ

Part II. True or false

Question 1. Consider the sphere

(S)\colon(x-1)^2+(y+2)^2+z^2=9

and the plane

(P)\colon 2x+y-2z+3=0

a) $(S)$ has centre $I(1, -2, 0)$ .	T / F
b) $(S)$ has radius $9$ .	T / F
c) The distance from the centre $I$ to $(P)$ is $1$ .	T / F
d) $(P)$ cuts $(S)$ in a circle of radius $2\sqrt2$ .	T / F

Part III. Short answer

Question 1. Find the distance between the parallel planes

(P)\colon 2x-y+2z+1=0

and

(Q)\colon 2x-y+2z-8=0

Question 2. In a room, choose axes

Oxyz

(unit: m) with the origin

O

at a corner of the floor. A light bulb hangs at the point

L(2, 3, 4)

. Find the distance from the bulb to the corner

O

(in m, to 2 decimal places).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $A(x-x_0)+B(y-y_0)+C(z-z_0)=0$ .

Steps:

Substitute $\vec n$ and the coordinates of $M$ into the formula.
Expand and simplify.

Exercise 2.

What you need: $d(M, (P))=\dfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$ .

Steps:

Compute the numerator $|2-4+3+3|$ .
Compute the denominator $\sqrt{4+4+1}$ .

Exercise 3.

What you need: $-2a=-2$ , $-2b=4$ , $-2c=-6$ ; $R=\sqrt{a^2+b^2+c^2-d}$ .

Steps:

Find $a$ , $b$ , $c$ .
Compute $R$ with $d=-2$ .

Exercise 4.

What you need: if $d\parallel(P)$ then $\vec u\cdot\vec n=0$ ; the intersection is found by substituting $x$ , $y$ , $z$ in terms of $t$ into $(P)$ .

Steps:

Read the direction vector from the coefficients of $t$ .
Compute $\vec u\cdot\vec n$ and solve the equation in $t$ .

Exercise 5.

What you need: $\cos\varphi=\dfrac{|\vec u_1\cdot\vec u_2|}{|\vec u_1|\cdot|\vec u_2|}$ .

Steps:

$\vec u_1=(1, 1, 0)$ and $\vec u_2=(0, 1, -1)$ .
Substitute into the formula.

Exercise 6.

What you need: the normal vector $\vec n=[\overrightarrow{AB}, \overrightarrow{AC}]$ .

Steps:

$\overrightarrow{AB}=(1, 1, -1)$ and $\overrightarrow{AC}=(-1, 2, 2)$ ; compute the vector product.
Write the equation of $(ABC)$ , then find the distance.

Exercise 7.

What you need: $AB=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}$ .

Steps:

Find $\overrightarrow{AB}$ .
Find its length.

Exercise 8.

What you need: the drone reaches the boundary of the zone when the coordinates of $M$ satisfy $x+2y+2z=30$ .

Steps:

Substitute the coordinates of $M$ into $x+2y+2z$ .
Solve the equation for $t$ .

Exercise 9.

What you need: $d$ takes $\vec n_P$ as its direction vector; $H=d\cap(P)$ .

Steps:

Write $d\colon x=3+t$ , $y=1+2t$ , $z=-2-2t$ .
Substitute into $(P)$ to find $t$ , hence $H$ ; then find $AH$ .

Exercise 10.

What you need: the radius is $R=d(I, (P))$ ; the point of contact is the orthogonal projection of $I$ on $(P)$ .

Steps:

Find $R$ .
Write the line through $I$ perpendicular to $(P)$ and find where it meets $(P)$ .

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