LemonmathSelf-study sheet

Grade 12 · Lesson 2.1

Vectors and their operations in space

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Goals

Recognise vectors in space, equal vectors and opposite vectors.
Add and subtract vectors with the triangle rule, the parallelogram rule and the parallelepiped rule.
Multiply a vector by a number and use the midpoint and centroid properties.
Find the angle and the scalar product (dot product) of two vectors, and use them to find lengths and angles and to prove perpendicularity.

Remember

Vectors, length, equal and opposite vectors in space are defined as in the plane.
The triangle and parallelogram rules still hold; add the parallelepiped rule $\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AA'}=\overrightarrow{AC'}$ .
Midpoint $I$ of $AB$ : $\overrightarrow{MA}+\overrightarrow{MB}=2\overrightarrow{MI}$ ; centroid $G$ : $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\vec{0}$ .
Angle between two vectors: draw them from a common point; the angle is from $0^\circ$ to $180^\circ$ .
$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos(\vec{a}, \vec{b})$ ; $\vec{a}^2=|\vec{a}|^2$ ; $\vec{a}\perp\vec{b}\iff\vec{a}\cdot\vec{b}=0$ .
To find a length: write the vector in terms of known edges, then square it.

Exercises

Exercise 1

Let $ABCD.A'B'C'D'$ be a parallelepiped. Simplify

a) $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD'}$ ; b) $\overrightarrow{AB}+\overrightarrow{A'D'}+\overrightarrow{CC'}$ .

Exercise 2

Let $ABCD$ be a tetrahedron. Simplify $\overrightarrow{AB}-\overrightarrow{AC}+\overrightarrow{DC}-\overrightarrow{DB}$ .

Exercise 3

Let $ABCD.A'B'C'D'$ be a cube with edge $3$ . Find $\overrightarrow{AB}\cdot\overrightarrow{AC}$ , $\overrightarrow{AB}\cdot\overrightarrow{A'C'}$ and $\overrightarrow{AA'}\cdot\overrightarrow{BD}$ .

Exercise 4

Let $ABCD$ be a regular tetrahedron with edge $2$ (all edges equal $2$ , all faces are equilateral triangles). Find $\overrightarrow{AB}\cdot\overrightarrow{AC}$ , then prove that $AB\perp CD$ .

Exercise 5

Three people pull a crate standing in the corner of a warehouse. The three pulling forces are pairwise perpendicular (along the two walls and straight up), with magnitudes $30$ N, $40$ N and $120$ N. Find the magnitude of the resultant force.

Exercise 6

Let $ABCD.A'B'C'D'$ be a cube with edge $1$ . Is each claim true or false?

a) $\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AA'}=\overrightarrow{AC'}$ .

b) $AC'=\sqrt3$ .

c) $\overrightarrow{AB}\cdot\overrightarrow{CD'}=1$ .

d) The angle between the vectors $\overrightarrow{AC}$ and $\overrightarrow{B'D'}$ is $90^\circ$ .

Exercise 7

A storeroom is a cuboid $ABCD.A'B'C'D'$ with $AB=4$ m, $AD=3$ m, $AA'=4$ m. Find the angle between the vectors $\overrightarrow{AC'}$ and $\overrightarrow{AC}$ (in degrees, rounded to the nearest whole number).

Exercise 8

(Revisiting Multiplying a vector by a number.) Let $ABCD$ be a tetrahedron and $G$ the centroid of triangle $BCD$ . Prove that $\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}=3\overrightarrow{AG}$ .

Exercise 9 Challenge

a) Prove that for any four points $A$ , $B$ , $C$ , $D$ : $\overrightarrow{AB}\cdot\overrightarrow{CD}+\overrightarrow{AC}\cdot\overrightarrow{DB}+\overrightarrow{AD}\cdot\overrightarrow{BC}=0.$ b) A tetrahedron $ABCD$ has $AB\perp CD$ and $AC\perp BD$ . Prove that $AD\perp BC$ .

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. Let

ABCD.A'B'C'D'

be a parallelepiped. The vector

\overrightarrow{AB}+\overrightarrow{AD}-\overrightarrow{CC'}

equals

\overrightarrow{AC}

.B.

\overrightarrow{AC'}

.C.

\overrightarrow{CA'}

.D.

\overrightarrow{A'C}

Question 2. Let

ABCD

be a tetrahedron, and

M

N

the midpoints of

AB

and

CD

. Then

\overrightarrow{AC}+\overrightarrow{BD}

equals

2\overrightarrow{MN}

.B.

\dfrac12\overrightarrow{MN}

.C.

\overrightarrow{MN}

.D.

-2\overrightarrow{MN}

Question 3. Let

ABCD.A'B'C'D'

be a cube. The angle between the vectors

\overrightarrow{AB}

and

\overrightarrow{C'A'}

45^\circ

.B.

135^\circ

.C.

90^\circ

.D.

60^\circ

Question 4. Let

ABCD

be a regular tetrahedron with edge

a

. The scalar product

\overrightarrow{AB}\cdot\overrightarrow{BC}

\dfrac{a^2\sqrt3}{2}

.B.

\dfrac{a^2}{2}

.C.

-\dfrac{a^2}{2}

.D.

0

Part II. True or false

Question 1. The cuboid

ABCD.A'B'C'D'

has

AB=2

AD=3

AA'=6

a) $\overrightarrow{AB}\cdot\overrightarrow{AA'}=0$ .	T / F
b) $AC'=7$ .	T / F
c) $\overrightarrow{AC'}\cdot\overrightarrow{AB}=2$ .	T / F
d) The angle between the vectors $\overrightarrow{AC'}$ and $\overrightarrow{AA'}$ is greater than $35^\circ$ .	T / F

Part III. Short answer

Question 1. Let

ABCD.A'B'C'D'

be a cube with edge

2

. Find

\overrightarrow{AC'}\cdot\overrightarrow{BD'}

Question 2. A drone takes off from a point

A

, flies

30

m east, then

40

m north, then straight up

20

m. How far is the drone's final position from

A

, in metres? Give your answer to 1 decimal place.

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

Hints (only when stuck)

Exercise 1.

What you need: the triangle rule; the parallelepiped rule; parallel edges in the same direction give equal vectors.

Steps:

a) The vectors already join tip to tail.
b) Replace $\overrightarrow{A'D'}=\overrightarrow{AD}$ and $\overrightarrow{CC'}=\overrightarrow{AA'}$ .

Exercise 2.

What you need: the subtraction rule $\overrightarrow{OX}-\overrightarrow{OY}=\overrightarrow{YX}$ .

Steps:

Combine $\overrightarrow{AB}-\overrightarrow{AC}$ .
Combine $\overrightarrow{DC}-\overrightarrow{DB}$ .

Exercise 3.

What you need: $\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos(\vec{a}, \vec{b})$ ; draw both vectors from a common point before measuring the angle.

Steps:

$AC=3\sqrt2$ and $\angle BAC=45^\circ$ .
$\overrightarrow{A'C'}=\overrightarrow{AC}$ .
$AA'$ is perpendicular to the base, so it is perpendicular to $BD$ .

Exercise 4.

What you need: the angles of an equilateral triangle are $60^\circ$ ; $\overrightarrow{CD}=\overrightarrow{AD}-\overrightarrow{AC}$ .

Steps:

Find $\overrightarrow{AB}\cdot\overrightarrow{AC}$ and $\overrightarrow{AB}\cdot\overrightarrow{AD}$ .
Find $\overrightarrow{AB}\cdot\overrightarrow{CD}$ by replacing $\overrightarrow{CD}$ .

Exercise 5.

What you need: the resultant is the sum of the three force vectors; for three pairwise perpendicular vectors, $|\vec{a}+\vec{b}+\vec{c}|^2=|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2$ .

Steps:

Square the sum of the three forces.
The scalar products of different forces are $0$ .

Exercise 6.

What you need: the parallelepiped rule; the three edges at $A$ are pairwise perpendicular with length $1$ ; write every vector in terms of $\overrightarrow{AB}$ , $\overrightarrow{AD}$ , $\overrightarrow{AA'}$ .

Steps:

b) Square $\overrightarrow{AC'}$ .
c) $\overrightarrow{CD'}=\overrightarrow{CD}+\overrightarrow{DD'}=-\overrightarrow{AB}+\overrightarrow{AA'}$ .
d) $\overrightarrow{B'D'}=\overrightarrow{BD}$ ; the diagonals of a square are perpendicular.

Exercise 7.

What you need: $\cos(\vec{a}, \vec{b})=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$ ; the parallelepiped rule.

Steps:

Write $\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{AD}$ and $\overrightarrow{AC'}=\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AA'}$ .
Find the scalar product and the two lengths.

Exercise 8.

What you need: $G$ is the centroid of triangle $BCD$ $\iff$ $\overrightarrow{GB}+\overrightarrow{GC}+\overrightarrow{GD}=\vec{0}$ (grade 10).

Steps:

Insert the point $G$ : $\overrightarrow{AB}=\overrightarrow{AG}+\overrightarrow{GB}$ , and so on.
Add the three equalities.

Exercise 9.

What you need: insert the point $A$ : $\overrightarrow{CD}=\overrightarrow{AD}-\overrightarrow{AC}$ ; the scalar product distributes over addition.

Steps:

Put $\vec{b}=\overrightarrow{AB}$ , $\vec{c}=\overrightarrow{AC}$ , $\vec{d}=\overrightarrow{AD}$ and write every vector in terms of them.
Expand; the terms cancel in pairs.
b) The first two products are $0$ .

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