LemonmathSelf-study sheet

Grade 12 · Lesson 2.2

Coordinate system in space

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Goals

Describe the coordinate system $Oxyz$ , its axes, its coordinate planes and the unit vectors $\vec{i}$ , $\vec{j}$ , $\vec{k}$ .
Read and write the coordinates of a point and of a vector; find the coordinates of $\overrightarrow{AB}$ .
Find the projection of a point onto the axes and the coordinate planes.
Find the distance between two points and the coordinates of a midpoint and a centroid.

Remember

$Oxyz$ : three pairwise perpendicular axes with unit vectors $\vec{i}$ , $\vec{j}$ , $\vec{k}$ of length $1$ , pairwise perpendicular.
$M(x, y, z)\iff\overrightarrow{OM}=x\vec{i}+y\vec{j}+z\vec{k}$ ; $\vec{a}=(x, y, z)\iff\vec{a}=x\vec{i}+y\vec{j}+z\vec{k}$ .
In $(Oxy)$ : $z=0$ ; on $Ox$ : $y=z=0$ . The projection of $(x, y, z)$ onto $(Oxy)$ is $(x, y, 0)$ .
$\overrightarrow{AB}=(x_B-x_A,\ y_B-y_A,\ z_B-z_A)$ ; $AB$ is the square root of the sum of three squares.
Midpoint, centroid: the mean of each coordinate.

Exercises

Exercise 1

a) Find the coordinates of the point $M$ given $\overrightarrow{OM}=2\vec{i}-3\vec{j}+\vec{k}$ .

b) Find the coordinates of the vector $\vec{a}=-\vec{i}+4\vec{k}$ .

Exercise 2

Let $N(4, -1, 3)$ . Find the projections of $N$ onto the planes $(Oxy)$ , $(Oyz)$ , $(Ozx)$ and onto the axis $Oy$ . Find the distance from $N$ to the plane $(Ozx)$ .

Exercise 3

Let $A(2, -1, 3)$ and $B(-1, 3, 3)$ . Find the coordinates of $\overrightarrow{AB}$ and the length $AB$ .

Exercise 4

Let $A(2, 1, -3)$ , $B(0, 5, 1)$ , $C(4, -3, 5)$ . Find the midpoint $M$ of $AB$ and the centroid $G$ of triangle $ABC$ .

Exercise 5

Let $A(1, -2, 3)$ and $M(2, 0, 1)$ . Find the point $N$ such that $M$ is the midpoint of $AN$ .

Exercise 6

A control station is at the origin $O$ of a system $Oxyz$ (unit: metre; $Ox$ points east, $Oy$ north, $Oz$ up). A drone is flying at the point $P(120, -50, 60)$ . How many metres is the drone from the station? (Round to the nearest whole number.)

Exercise 7

Let $M(3, -4, 12)$ . Is each claim true or false?

a) The projection of $M$ onto the plane $(Oxy)$ is $(3, -4, 0)$ .

b) The distance from $M$ to the plane $(Oyz)$ is $4$ .

c) $OM=13$ .

d) The distance from $M$ to the axis $Ox$ is $4$ .

Exercise 8

The cube $ABCD.A'B'C'D'$ with edge $4$ is placed in $Oxyz$ so that $A$ is at $O$ , $B(4, 0, 0)$ , $D(0, 4, 0)$ , $A'(0, 0, 4)$ . Find the coordinates of $C'$ , of the centre $I$ of the face $A'B'C'D'$ , and the length $IB$ .

Exercise 9

(Revisiting Vectors in the coordinate plane.) In the plane $Oxy$ , let $A(1, 2)$ and $B(4, 6)$ .

a) Find $AB$ with the grade-10 formula.

b) Regard $A$ , $B$ as the points $A(1, 2, 0)$ , $B(4, 6, 0)$ of the plane $(Oxy)$ in space. Lift $B$ straight up to $B'(4, 6, 12)$ . Find $AB'$ .

Exercise 10 Challenge

Find the point $M$ on the axis $Oz$ that is equidistant from $A(1, 2, 3)$ and $B(-1, 0, 5)$ .

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 the system

Oxyz

, the point

M

satisfies

\overrightarrow{OM}=3\vec{i}-\vec{k}

. The coordinates of

M

are

(0, 3, -1)

.B.

(3, -1, 0)

.C.

(3, 0, -1)

.D.

(3, 0, 1)

Question 2. The orthogonal projection of the point

A(2, -5, 4)

onto the plane

(Oyz)

(2, 0, 0)

.B.

(0, -5, 4)

.C.

(2, -5, 0)

.D.

(2, 0, 4)

Question 3. Let

A(1, -2, 3)

and

B(3, 0, -1)

. The midpoint

I

of the segment

AB

I(2, 2, -4)

.B.

I(1, 1, -2)

.C.

I(4, -2, 2)

.D.

I(2, -1, 1)

Question 4. The distance from the point

M(3, -4, 6)

to the axis

Oz

5

.B.

6

.C.

\sqrt{61}

.D.

\sqrt{52}

Part II. True or false

Question 1. In the system

Oxyz

, let

A(1, 2, -1)

B(3, -2, 3)

C(-1, 3, 1)

a) $\overrightarrow{AB}=(2, -4, 4)$ .	T / F
b) $AB=6$ .	T / F
c) The centroid of triangle $ABC$ is $G(3, 3, 3)$ .	T / F
d) The point $D$ that makes $ABCD$ a parallelogram is $D(-3, 7, -3)$ .	T / F

Part III. Short answer

Question 1. Let

A(1, 0, 0)

B(0, 2, 0)

C(0, 0, 2)

. Find the perimeter of triangle

ABC

, giving your answer to 1 decimal place.

Question 2. In a system

Oxyz

attached to a training ground (unit: metre), two drones are at the points

P(10, 20, 15)

and

Q(40, -20, 45)

. How far apart are the drones, in metres? Give your answer to 1 decimal place.

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

Hints (only when stuck)

Exercise 1.

What you need: $\overrightarrow{OM}=x\vec{i}+y\vec{j}+z\vec{k}\iff M(x, y, z)$ ; a missing component has coefficient $0$ .

Steps:

Read off the coefficients of $\vec{i}$ , $\vec{j}$ , $\vec{k}$ in turn.
b) The coefficient of $\vec{j}$ is $0$ .

Exercise 2.

What you need: projection onto a coordinate plane: set the remaining coordinate to $0$ ; the plane $(Ozx)$ is $y=0$ .

Steps:

Write the three projections onto the three planes.
Onto the axis $Oy$ : keep only the y-coordinate.
The distance to $(Ozx)$ is $|y|$ .

Exercise 3.

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

Steps:

Subtract coordinate by coordinate.
Add the three squares and take the square root.

Exercise 4.

What you need: midpoint and centroid: the mean of each coordinate.

Steps:

Add each coordinate of $A$ and $B$ , then divide by $2$ .
Add each coordinate of $A$ , $B$ , $C$ , then divide by $3$ .

Exercise 5.

What you need: $M$ is the midpoint of $AN$ $\iff$ $x_M=\dfrac{x_A+x_N}{2}$ (and likewise), that is $x_N=2x_M-x_A$ .

Steps:

Double the coordinates of $M$ .
Subtract the coordinates of $A$ .

Exercise 6.

What you need: $OP=\sqrt{x^2+y^2+z^2}$ .

Steps:

Square the three coordinates and add them.
Take the square root and round.

Exercise 7.

What you need: the distance to $(Oyz)$ is $|x|$ ; the distance to the axis $Ox$ is $\sqrt{y^2+z^2}$ .

Steps:

Write the projection onto $(Oxy)$ .
Find $OM$ from the three coordinates.
Decide which coordinates each distance uses.

Exercise 8.

What you need: the parallelepiped rule $\overrightarrow{AC'}=\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AA'}$ ; the centre of a square is the midpoint of a diagonal.

Steps:

Add the three edge vectors to get $C'$ .
$I$ is the midpoint of $A'C'$ .
Find $IB$ with the distance formula.

Exercise 9.

What you need: $AB=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}$ in the plane; in space there is an extra $(z_B-z_A)^2$ .

Steps:

a) Find the two coordinate differences.
b) Add the height difference $12$ and notice that $ABB'$ has a right angle at $B$ .

Exercise 10.

What you need: a point on $Oz$ has the form $M(0, 0, t)$ ; $MA=MB\iff MA^2=MB^2$ .

Steps:

Write $MA^2$ and $MB^2$ in terms of $t$ .
Set them equal; the $t^2$ terms cancel.

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