Wize High School Grade 12 Calculus Textbook > Vectors

Introduction to Vectors

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What are Points?
   How does a GPS work? How would you describe the position of a treasure on a map?
We can represent any position in 2D using the Cartesian coordinates (x, y)\left(x,\ y\right)(x, y)

Wize Concept
The 2D or R2R^2R2 space contains the xxx and yyy axes.

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  How could we describe the position of a plane in the sky?
We can represent any position in 3D using the coordinates (x, y, z)\left(x,\ y,\ z\right)(x, y, z)
❓ How do we know which directions are the positive axes pointing in?
Use the right-hand rule!
In R3R^3R3 we have the following special planes:
The xyxyxy-plane contains all points where z=0
The xzxzxz-plane contains all points where y=0
The yzyzyz-plane contains all points where x=0
Wize Concept
The 3D or R3R^3R3 space contains the xxx, yyy, and zzz axes.

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What are Scalars?
A scalar is a quantity with magnitude only  →  it is just a numerical number!

Examples
Distance
: The library is 2 km from your house
Speed
: The car is travelling at 110 km/h
Length/height
: LeBron is 206 cm tall
Mass
: Peter Griffin is 270 lbs

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What are Vectors?
If you had to tell your friend where a treasure is, we can give them the coordinates of the treasure, or you can give them the directions to the treasure!
   
Vectors allow us to describe a quantity that involves both magnitude and direction → a number and a direction!

Examples
Displacement
: The library is 2 km N30oW from your home
Velocity
: The car is travelling at 110 km/h E
Force
: We need to apply a force of 50 N 25o above the ground to move the box
Acceleration
: The car accelerates at a rate of 10 m/s210\ m/s^210 m/s2 W (the velocity is increasing constantly)

Practice: Scalar VS Vector?
Each of the following is either a scalar or a vector quantity.
Select all of the vector quantities in the list.

The mass of a newborn puppy
[Hint: mass is usually measured in lb or kg]

The temperature in Ontario on April 1st
[Hint: temperature is usually measured in oC or oF]

The weight of an apple
[Hint: weight is the measure of force downwards due to gravity]

The radius of a circle

The acceleration of a car
[Hint: acceleration is the measure of the change in velocity]

The density of a metal
[Hint: density is the measure of mass per volume]

The volume of a hot air balloon

The momentum of a train
[Hint: momentum is calculated by mass ×velocitymass\ \times velocitymass ×velocity]

I don't know

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2D Vectors
Cartesian Form of a Vector
A vector in 2D can be written as v⃗=[v1, v2]\vec v=[v_1,\ v_2]v=[v1​, v2​] where
v1v_1v1​ is the x\color{orange}xx-component of the vector (number of "steps" in the xxx-direction)
v2v_2v2​ is the y\color{orange}yy-component of the vector (number of "steps" in the yyy-direction)
Example

v⃗=[−2,−1]\vec v=\begin{bmatrix}
-2, -1
\end{bmatrix}v=[−2,−1​]

Graphical/geometrical representation of a vector
A vector in 2D can be drawn as a directed line segment (or ray)

Example

v⃗=[−2, −1]\vec{v}=\left[-2,\ -1\right]v=[−2, −1] is the directed line segment that goes 2 units in the negative x direction (left) and 1 unit in the negative y direction (down)

Polar Form of a Vector
A vector in 2D can be described as the magnitude rrr and direction θ\thetaθ

Example

v⃗=[−2, −1]\vec{v}=\left[-2,\ -1\right]v=[−2, −1] is a vector with length 5\sqrt{5}5​ and angle 206.6°206.6\degree206.6° measured counter clockwise from the positive x-axis

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Definitions w/ Vectors
Equal vectors are vectors that have the same magnitude and directions.

Parallel/collinear vectors are scalar multiples of one another.
Example: v⃗=[2, 1]\vec v=[2,\ 1]v=[2, 1] and u⃗=[6, 3]\vec u=[6,\ 3]u=[6, 3] are parallel

Opposite vectors are negative scalar multiples of one another.
Example: v⃗=[2, −1]\vec v=[2,\ -1]v=[2, −1] and u⃗=[−4, 2]\vec u=
[-4,\ 2]u=[−4, 2] are opposite

Position vector
AB→\overrightarrow{AB}AB is a position vector that points from point AAA to point BBB.
Example: Draw the position vector that points from the point A(1,−2)A(1,-2)A(1,−2) to the point B(2,2)B(2, 2)B(2,2)

Wize Tip
We can label the vector AB→\overrightarrow{AB}AB with a single lower case letter u⃗\vec uu.

Practice: 2D Vectors
Given the diagram below, answer the following questions.

Which of the following vector(s) is/are the same (equal) as AB→\overrightarrow{AB}AB?

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Polar  →  Cartesian Form
Suppose we are given the Polar form of a vector with length rrr and angle θ\thetaθ measured counter clockwise from the positive xxx-axis, we can use trigonometric relationships to convert it to Cartesian form.

Write it Down
x=rcos⁡θx=r\cos\thetax=rcosθ
y=rsin⁡θy=r\sin\thetay=rsinθ

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Cartesian Form  →  Polar Form
Suppose we are given the Cartesian form of a vector v⃗=[x, y]\vec{v}=[x,\ y]v=[x, y], we can use trigonometric relationships to convert it to polar form.

Write it Down
r=x2+y2r=\sqrt{x^2+y^2}r=x2+y2​
θ=tan⁡−1(yx)\displaystyle \theta=\tan^{-1}\left(\displaystyle \frac{y}{x}\right)θ=tan−1(xy​)   *measured from the x-axis

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Did you Know?
There are many ways to represent directions (a.k.a bearings). Here are a few common ones that all mean the same thing!

Don't let these special angles trick you!

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Example: Converting Between Vector Forms
1) Convert the vector v⃗=[5, −12]\vec{v}=\left[5,\ -12\right]v=[5, −12] into polar form.

r=(5)2+(−12)2=169=13r=\sqrt{\left(5\right)^2+\left(-12\right)^2}=\sqrt{169}=13r=(5)2+(−12)2​=169​=13
θ=tan⁡−1(−125)≈−67.38°\theta=\tan^{-1}\left(-\frac{12}{5}\right)\approx-67.38\degreeθ=tan−1(−512​)≈−67.38° (since this angle is negative, it means that it is measured clockwise from the positive x-axis)
Therefore, this vector has a magnitude of 13 and is 67.38o measured clockwise from the positive x-axis.

2) Convert the vector with magnitude 101010 and forms a 150°150\degree150° angle counter clockwise from the positive xxx-axis.

x=10cos⁡(150°)=−10(32)=−53x=10\cos\left(150\degree\right)=-10\left(\frac{\sqrt{3}}{2}\right)=-5\sqrt{3}x=10cos(150°)=−10(23​​)=−53​
y=10sin⁡(150°)=10(12)=5y=10\sin\left(150\degree\right)=10\left(\frac{1}{2}\right)=5y=10sin(150°)=10(21​)=5
Therefore, this vector in Cartesian form is [−53, 5]\left[-5\sqrt{3},\ 5\right][−53​, 5]

Practice: Cartesian  →  Polar
The velocity of a car is described by the vector [−30, 40]\left[-30,\ 40\right][−30, 40]km/h. 

Describe the speed (magnitude) and direction of this car.

The magnitude of the vector = _________ km/h

The angle measured counter clockwise from the positive x-axis = ____________

\degree

(round to 1 decimal place)

I don't know

Practice: Polar  →  Cartesian
In a car accident involving a Honda Civic and a Ford, the Civic's momentum is 65200 kg m/s N25oW. Write this vector in Cartesian form [x, y]\left[x,\ y\right][x, y]

x=

(round to 1 decimal place)

y=

(round to 1 decimal place)

I don't know

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3D Vectors
Cartesian Form of a Vector
A vector in 3D can be written as v⃗=[v1, v2, v3]\vec{v}=[v_1,\ v_2,\ v_3]v=[v1​, v2​, v3​] where
v1v_1v1​ is the x\color{orange}xx-component of the vector (number of "steps" in the xxx-direction)
v2v_2v2​ is the y\color{orange}yy-component of the vector (number of "steps" in the yyy-direction)
v3v_3v3​ is the z\color{orange}zz-component of the vector (number of "steps" in the zzz-direction)
Example

v⃗=[300, 200, 1000]\vec v=\begin{bmatrix}
300,\ 200,\ 1000
\end{bmatrix}v=[300, 200, 1000​]

Graphical/geometrical representation of a vector
A vector in 3D can be drawn as a directed line segment (or ray)

Example

Wize Tip
The definitions for parallel, opposite, and position vectors in 3D are the same as those in 2D.

Watch Out!
Unlike vectors in 2D, we cannot represent 3D vectors with just a magnitude and angle, we actually need 3 angles -- one for each of the axes.