LemonmathSelf-study sheet

Grade 12 · Lesson 2.3

Operations on vectors in coordinates

Name: Date: Time spent: min

Goals

Add, subtract and multiply vectors by a number using coordinates in space.
Compute the scalar product (dot product) $\vec{a}\cdot\vec{b}=x_1x_2+y_1y_2+z_1z_2$ and the length $|\vec{a}|$ .
Find the angle between two vectors; recognise perpendicular and collinear vectors.
Use coordinates to find a vertex of a parallelogram, an angle of a triangle, and to prove perpendicularity.

Remember

Add, subtract, multiply by a number: work on each coordinate.
$\vec{b}$ is collinear with $\vec{a}\neq\vec{0}$ $\iff$ $\vec{b}=k\vec{a}$ ; $A$ , $B$ , $C$ are collinear $\iff$ $\overrightarrow{AB}$ , $\overrightarrow{AC}$ are collinear.
$\vec{a}\cdot\vec{b}=x_1x_2+y_1y_2+z_1z_2$ (a number); $|\vec{a}|=\sqrt{x_1^2+y_1^2+z_1^2}$ .
$\cos(\vec{a}, \vec{b})=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$ ; $\vec{a}\perp\vec{b}\iff\vec{a}\cdot\vec{b}=0$ .
The angle $\angle BAC$ of a triangle is the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .
Parallelogram $ABCD$ : $\overrightarrow{AD}=\overrightarrow{BC}$ .

Exercises

Exercise 1

Let $\vec{u}=(2, -1, 3)$ and $\vec{v}=(-1, 4, 0)$ . Find $\vec{u}+\vec{v}$ , $3\vec{u}-2\vec{v}$ and $\vec{u}\cdot\vec{v}$ .

Exercise 2

Let $\vec{a}=(2, -3, 6)$ . Find $|\vec{a}|$ and the vector $\vec{e}$ with the same direction as $\vec{a}$ and length $1$ .

Exercise 3

Find the angle between the vectors $\vec{a}=(1, 1, 0)$ and $\vec{b}=(0, 1, 1)$ .

Exercise 4

Among the three vectors $\vec{u}=(1, 2, -1)$ , $\vec{v}=(3, -1, 1)$ , $\vec{w}=(1, 1, 3)$ , which pairs are perpendicular?

Exercise 5

Find $x$ , $y$ such that the vectors $\vec{a}=(1, x, 3)$ and $\vec{b}=(-2, 4, y)$ are collinear.

Exercise 6

Let $A(3, 2, -1)$ , $B(1, 1, 1)$ , $C(2, 3, 3)$ . Prove that triangle $ABC$ has a right angle at $B$ and find its area.

Exercise 7

Let $\vec{a}=(1, -2, 2)$ and $\vec{b}=(2, 2, 1)$ . Is each claim true or false?

a) $|\vec{a}|=3$ .

b) $\vec{a}\perp\vec{b}$ .

c) $\vec{a}-\vec{b}=(-1, -4, 3)$ .

d) $|\vec{a}+\vec{b}|=6$ .

Exercise 8

A person pulls a sledge with the force $\vec{F}=(10, 5, -2)$ (unit: N). The sledge moves along the displacement $\vec{d}=(3, 4, 0)$ (unit: m). Find the angle between the pulling force and the direction of motion (in degrees, rounded to the nearest whole number).

Exercise 9

(Revisiting Vectors and their operations in space.) In the previous lesson, by decomposing vectors, we found that the angle between $\overrightarrow{AC}$ and $\overrightarrow{A'D}$ in the cube $ABCD.A'B'C'D'$ is $60^\circ$ . Find it again with coordinates, for the cube with edge $1$ , $A$ at the origin, $B(1, 0, 0)$ , $D(0, 1, 0)$ , $A'(0, 0, 1)$ .

Exercise 10 Challenge

Let $A(2, 1, 1)$ and $B(4, -1, 1)$ . Find the points $M$ on the axis $Ox$ such that triangle $MAB$ has a right angle at $M$ .

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

, let

\vec{a}=(2, -1, 3)

and

\vec{b}=(1, 0, -2)

. The coordinates of

\vec{a}-2\vec{b}

are

(0, -1, -1)

.B.

(0, -1, 7)

.C.

(4, -1, -1)

.D.

(1, -1, 5)

Question 2. Let

\vec{u}=(3, -2, 1)

and

\vec{v}=(1, 4, -2)

. The scalar product

\vec{u}\cdot\vec{v}

7

.B.

(3, -8, -2)

.C.

-7

.D.

-3

Question 3. Which vector is perpendicular to

\vec{a}=(1, 2, -2)

(1, 2, 2)

.B.

(2, -1, 2)

.C.

(-2, -4, 4)

.D.

(2, 1, 2)

Question 4. The angle between the vectors

\vec{a}=(1, 0, 1)

and

\vec{b}=(-1, 1, 0)

120^\circ

.B.

60^\circ

.C.

45^\circ

.D.

150^\circ

Part II. True or false

Question 1. In the system

Oxyz

, let

A(1, 0, 2)

B(3, 1, 0)

C(0, 2, 2)

a) $\overrightarrow{AB}=(2, 1, -2)$ .	T / F
b) Triangle $ABC$ has a right angle at $A$ .	T / F
c) The area of triangle $ABC$ is $3\sqrt5$ .	T / F
d) The angle $\angle ABC$ is greater than $30^\circ$ .	T / F

Part III. Short answer

Question 1. Let

A(1, 2, -1)

B(3, 0, 1)

and

C(x, y, 5)

. Given that

A

B

C

are collinear, find

x+y

Question 2. A force

\vec{F}=(4, -2, 3)

(unit: N) moves an object from

M(1, 0, 0)

N(5, 3, 2)

(unit: m). Find the angle between the force

\vec{F}

and the direction of motion

\overrightarrow{MN}

, in degrees, to 1 decimal place.

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

Hints (only when stuck)

Exercise 1.

What you need: add, subtract and multiply by a number coordinate by coordinate; $\vec{u}\cdot\vec{v}=x_1x_2+y_1y_2+z_1z_2$ .

Steps:

Compute $3\vec{u}$ and $2\vec{v}$ first.
Multiply coordinates in the same position and add.

Exercise 2.

What you need: $|\vec{a}|=\sqrt{x^2+y^2+z^2}$ ; $\dfrac{1}{|\vec{a}|}\vec{a}$ has the same direction as $\vec{a}$ and length $1$ .

Steps:

Compute $|\vec{a}|$ .
Divide each coordinate by $|\vec{a}|$ .

Exercise 3.

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

Steps:

Compute $\vec{a}\cdot\vec{b}$ , $|\vec{a}|$ , $|\vec{b}|$ .
Find the angle with that cosine.

Exercise 4.

What you need: $\vec{a}\perp\vec{b}\iff\vec{a}\cdot\vec{b}=0$ .

Steps:

Compute the three scalar products $\vec{u}\cdot\vec{v}$ , $\vec{u}\cdot\vec{w}$ , $\vec{v}\cdot\vec{w}$ .
Pick those equal to $0$ .

Exercise 5.

What you need: $\vec{b}$ is collinear with $\vec{a}$ $\iff$ $\vec{b}=k\vec{a}$ .

Steps:

Find $k$ from the x-coordinates: $-2=k\cdot1$ .
Use $k$ for the other two coordinates.

Exercise 6.

What you need: right angle at $B$ $\iff$ $\overrightarrow{BA}\cdot\overrightarrow{BC}=0$ ; the area of a right-angled triangle is half the product of the two shorter sides.

Steps:

Find $\overrightarrow{BA}$ , $\overrightarrow{BC}$ and their scalar product.
Find $BA$ , $BC$ .

Exercise 7.

What you need: the formulas for length, scalar product and subtraction in coordinates.

Steps:

Compute $|\vec{a}|$ and $\vec{a}\cdot\vec{b}$ .
Subtract coordinate by coordinate.
Compute $\vec{a}+\vec{b}$ and its length.

Exercise 8.

What you need: $\cos(\vec{F}, \vec{d})=\dfrac{\vec{F}\cdot\vec{d}}{|\vec{F}||\vec{d}|}$ ; in physics, $\vec{F}\cdot\vec{d}$ is the work done by the force (unit: J).

Steps:

Compute $\vec{F}\cdot\vec{d}$ .
Compute $|\vec{F}|$ , $|\vec{d}|$ and the cosine of the angle.

Exercise 9.

What you need: the coordinates of the vertices of the cube; the cosine formula for the angle.

Steps:

Write $C(1, 1, 0)$ and find $\overrightarrow{AC}$ , $\overrightarrow{A'D}$ .
Compute the cosine.

Exercise 10.

What you need: a point on $Ox$ has the form $M(t, 0, 0)$ ; right angle at $M$ $\iff$ $\overrightarrow{MA}\cdot\overrightarrow{MB}=0$ .

Steps:

Write $\overrightarrow{MA}$ , $\overrightarrow{MB}$ in terms of $t$ .
Set the scalar product equal to $0$ and solve.

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