Vector Basics Slides Unit 01: Vector Analysis
Defining Vectors
Introduction to magnitude, direction, and geometric representation.
ENG-SPEC-001
The Treasure Hunt Dilemma
"Walk 50 paces. Then walk 30 paces. Dig there."
If you follow these instructions, where do you end up? Is there only one possible location?
Scalar (Size only)
"50 paces"
Vector (Size + Dir)
"50 paces North"
Without Direction ,Undefined .
Scalar vs. Vector
Scalar Quantities
Quantities described by magnitude (size) alone.
Mass (15 kg)
Temperature (22°C)
Time (5 seconds)
Speed (60 mph)
Vector Quantities
Quantities described by both magnitude AND direction.
Displacement (10m East)
Velocity (60 mph North)
Force (5N Downward)
Acceleration (9.8 m/s² West)
Geometric Representation
Vectors are represented by directed line segments (arrows).
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Length = Magnitude
The physical length of the arrow represents the amount.
2
Arrowhead = Direction
Points where the quantity is heading.
Notation
\[ \vec{v} \quad \text{or} \quad \mathbf{v} \quad \text{or} \quad \vec{AB} \]
Initial Point (A) Terminal Point (B) v
Magnitude & Direction
Magnitude
The distance between the initial and terminal points. Always non-negative.
\[ ||\vec{v}|| \]
Direction
The angle \(\theta\) the vector makes with a reference line (usually the positive x-axis).
\[ \theta = 45^\circ \text{ or } \text{bearing } 090^\circ \]
Note: Two vectors are equal if and only if they have the same magnitude AND the same direction, regardless of where they start!
Vector Identification Worksheet Vector Identification
Lesson 1: Scalars, Vectors, and Geometric Forms
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1
Scalar vs. Vector Analysis
Determine whether each quantity is a scalar or a vector. Briefly explain your reasoning.
A. An airplane flying at 450 km/h due East.
Scalar
Vector
B. A rock with a mass of 12 kilograms.
Scalar
Vector
C. Pushing a crate with 50 Newtons of force downwards.
Scalar
Vector
2
Drawing Directed Line Segments
On the grid below, draw the following vectors starting from the given initial points. Label each vector.
Vector \(\vec{u}\)
Initial Point: \((1, 1)\)
Terminal Point: \((4, 5)\)
Vector \(\vec{v}\)
Initial Point: \((6, 8)\)
Go 3 units Left, 4 units Down
Vector \(\vec{w}\)
Initial Point: \((2, 7)\)
Terminal Point: \((2, 2)\)
X
Y
3
Critical Thinking
1. Can a vector have a magnitude of 0? If so, what is its direction? Explain your reasoning.
2. If vector \(\vec{A}\) and vector \(\vec{B}\) have the same magnitude, are they necessarily equal? Use a sketch to support your answer.
Sketch Area
Vector Identification Answer Key Vector Identification [Answer Key]
Lesson 1: Teacher Facilitation Guide
Instructor Copy
1. Scalar vs. Vector Analysis
A. Airplane flying at 450 km/h due East.
Vector
Reason: Includes both magnitude (450 km/h) and direction (East).
B. A rock with a mass of 12 kilograms.
Scalar
Reason: Mass only has magnitude; it has no spatial direction.
C. Pushing a crate with 50 Newtons of force downwards.
Vector
Reason: Force includes magnitude (50N) and direction (downwards).
2. Drawing Directed Line Segments
\(\vec{u}\): (1,1) to (4,5)
$\Delta x = 3, \Delta y = 4$. Magnitude = 5.
\(\vec{v}\): (6,8) to (3,4)
Terminal point is (3,4) because of "3 left, 4 down".
\(\vec{w}\): (2,7) to (2,2)
Vertical vector pointing straight down.
u v w
3. Critical Thinking
1. Can a vector have a magnitude of 0?
Yes, this is called the Zero Vector (\(\vec{0}\)). Its magnitude is 0, but unlike other vectors, its direction is undefined or indeterminate because it has no length to point in any specific direction.
2. If vectors have same magnitude, are they equal?
No. Equality requires both magnitude AND direction to be the same. Two vectors could both be 5 units long, but if one points North and the other East, they are not equal.
Geometric Addition Slides Lesson 02
Vector Operations
Geometric addition, subtraction, and the resultant vector.
What is a Resultant?
When multiple vectors act on an object, the **Resultant Vector** is the single vector that represents their combined effect.
\[ \vec{R} = \vec{A} + \vec{B} \]
Think of it as the "net" outcome of two different movements or forces.
Example: A plane flying North while the wind blows East. The **resultant** is the actual path the plane travels over the ground.
Triangle Law (Tip-to-Tail)
Step 1
Draw the first vector (\(\vec{A}\)).
Step 2
Draw the second vector (\(\vec{B}\)) starting from the tip of \(\vec{A}\).
Step 3
The resultant (\(\vec{A}+\vec{B}\)) goes from the tail of \(\vec{A}\) to the tip of \(\vec{B}\).
A B A + B
Parallelogram Law (Tail-to-Tail)
When vectors start from the same point:
Resultant
Vector Subtraction
Subtracting a vector is the same as adding its opposite .
\[ \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) \]
The vector \(-\vec{B}\) has the same magnitude as \(\vec{B}\) but points in the exactly opposite direction .
B -B
Visualizing Sums Worksheet Visualizing Sums
Lesson 2: Geometric Vector Addition & Subtraction
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The Triangle Method (Tip-to-Tail)
Given the vectors \(\vec{a}\) and \(\vec{b}\) below, use the tip-to-tail method to draw the resultant vector \(\vec{a} + \vec{b}\). Label the resultant clearly.
Givens
a b
Workspace: Triangle Method
2
The Parallelogram Method
Using the same vectors starting from a common origin, draw the resultant vector \(\vec{a} + \vec{b}\). Show the dashed lines used to complete the parallelogram.
Workspace: Parallelogram Method
3
Vector Subtraction
Draw the resultant vector for \(\vec{a} - \vec{b}\). Remember: \(\vec{a} - \vec{b} = \vec{a} + (-\vec{b})\).
Workspace: Subtraction
4
Equilibrium Challenge
An object is being pulled by two forces: Force \(\vec{F}_1\) (30N East) and Force \(\vec{F}_2\) (40N North). Draw a third force vector, \(\vec{F}_{eq}\), that would keep the object in **equilibrium** (Net force = 0).
Visualizing Sums Answer Key Visualizing Sums [Answer Key]
Lesson 2: Geometric Vector Addition & Subtraction
Instructor Guide
1. Triangle Method
a + b
Tip of \(\vec{a}\) connects to tail of \(\vec{b}\). Resultant spans from initial tail to final tip.
2. Parallelogram Method
Start both at same point. Complete the box. Resultant is the diagonal from the origin.
3. Subtraction (a - b)
-b
Reverse the direction of \(\vec{b}\) to get \(-\vec{b}\). Then perform standard addition.
4. Equilibrium
F1 (30N E) F2 (40N N) F_eq (50N SW)
The equilibrium force \(\vec{F}_{eq}\) must be equal in magnitude but opposite in direction to the resultant of \(\vec{F}_1\) and \(\vec{F}_2\).
Common Misconception Alert
Students often forget that vector addition is commutative (\(\vec{a}+\vec{b} = \vec{b}+\vec{a}\)). Encourage them to draw it both ways to see that they end up at the same terminal point regardless of the order.
Component Form Slides Lesson 03
Component Form
Transitioning from geometric segments to algebraic coordinate pairs.
CARTESIAN_SYSTEM_v3.0
The Power of Algebra
Drawing vectors is helpful for visualization, but algebra is necessary for precision and calculation.
"Instead of drawing a 4-cm line at 30 degrees, we describe it by how much it moves Right and how much it moves Up."
Coordinate Mapping
\[ \vec{v} = \langle x, y \rangle \]
x-component y-component
Calculating Components
If a vector starts at \((x_1, y_1)\) and ends at \((x_2, y_2)\), its component form is found by:
\[ \vec{v} = \langle x_2 - x_1, \ y_2 - y_1 \rangle \]
Standard Position: A vector is in standard position if its initial point is at the origin \((0, 0)\).
Example
Initial: \(A(2, 3)\)
Terminal: \(B(5, -1)\)
\[ \vec{AB} = \langle 5-2, \ -1-3 \rangle = \langle 3, -4 \rangle \]
Magnitude Formula
The magnitude of a vector is simply the length of the segment, found using the Pythagorean Theorem .
\[ ||\vec{v}|| = \sqrt{x^2 + y^2} \]
Master It
Find magnitude of \(\vec{u} = \langle 8, 15 \rangle\):
\[ ||\vec{u}|| = \sqrt{8^2 + 15^2} \]
\[ ||\vec{u}|| = \sqrt{64 + 225} \]
\[ ||\vec{u}|| = 17 \]
Unit Vectors \(\mathbf{i}\) and \(\mathbf{j}\)
Standard unit vectors have a magnitude of exactly 1 and point along the axes.
\(\mathbf{i}\)
\( \langle 1, 0 \rangle \)
\(\mathbf{j}\)
\( \langle 0, 1 \rangle \)
Linear Combination Form
Any vector can be written as a sum of these unit vectors:
\[ \langle a, b \rangle = a\mathbf{i} + b\mathbf{j} \]
Example: \( \langle 3, -5 \rangle = 3\mathbf{i} - 5\mathbf{j} \)
Resolution & Magnitude Worksheet Resolution & Magnitude
Lesson 3: Component Form & Unit Vector Notation
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Finding Component Form
Find the component form \(\langle x, y \rangle\) for each vector with the given initial and terminal points.
A. Initial: \((3, 2)\), Terminal: \((7, 5)\)
Ans: \(\langle \ \ \ , \ \ \ \rangle\)
B. Initial: \((0, 0)\), Terminal: \((-4, 6)\)
Ans: \(\langle \ \ \ , \ \ \ \rangle\)
C. Initial: \((5, -1)\), Terminal: \((1, -4)\)
Ans: \(\langle \ \ \ , \ \ \ \rangle\)
D. Initial: \((-2, -3)\), Terminal: \((3, 9)\)
Ans: \(\langle \ \ \ , \ \ \ \rangle\)
2
Calculating Magnitude
Calculate the magnitude \(||\vec{v}||\) of each vector. Show your work using the Pythagorean theorem. Leave answers in simplest radical form if necessary.
\[ \vec{u} = \langle 3, 4 \rangle \]
Show Work
\[ \vec{v} = \langle -5, 12 \rangle \]
Show Work
\[ \vec{w} = \langle 6, -2 \rangle \]
Show Work
3
Unit Vector Notation
Convert between component form \(\langle a, b \rangle\) and unit vector form \(a\mathbf{i} + b\mathbf{j}\).
\[ \langle 7, -2 \rangle \]
\[ 3\mathbf{i} + 4\mathbf{j} \]
\[ \langle 0, 5 \rangle \]
\[ -\mathbf{i} - \mathbf{j} \]
4
Movement Scripting
Script Header: Player_Movement.js
PLAYER_START = (100, 100);
MOVE_TO = (250, 400);
// TASK: Calculate the vector \(\vec{m}\) required to perform this move.
\(\vec{m} = \langle \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \rangle\)
// TASK: Calculate the total distance of this move.
Distance = _________________
Component Form Answer Key Resolution & Magnitude [Answer Key]
Lesson 3: Component Form & Unit Vector Notation
Instructor Copy
1. Finding Component Form
A. (3, 2) to (7, 5)
\[ \langle 7-3, 5-2 \rangle = \langle 4, 3 \rangle \]
B. (0, 0) to (-4, 6)
\[ \langle -4-0, 6-0 \rangle = \langle -4, 6 \rangle \]
C. (5, -1) to (1, -4)
\[ \langle 1-5, -4-(-1) \rangle = \langle -4, -3 \rangle \]
D. (-2, -3) to (3, 9)
\[ \langle 3-(-2), 9-(-3) \rangle = \langle 5, 12 \rangle \]
2. Calculating Magnitude
\( \vec{u} = \langle 3, 4 \rangle \)
\( \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
\( \vec{v} = \langle -5, 12 \rangle \)
\( \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \)
\( \vec{w} = \langle 6, -2 \rangle \)
\( \sqrt{6^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \)
3. Unit Vector Notation
\( \langle 7, -2 \rangle \) \( 7\mathbf{i} - 2\mathbf{j} \)
\( 3\mathbf{i} + 4\mathbf{j} \) \( \langle 3, 4 \rangle \)
\( \langle 0, 5 \rangle \) \( 5\mathbf{j} \)
\( -\mathbf{i} - \mathbf{j} \) \( \langle -1, -1 \rangle \)
4. Movement Scripting
// SOLUTIONS
Vector \(\vec{m} = \langle 250 - 100, 400 - 100 \rangle = \mathbf{\langle 150, 300 \rangle}\)
Distance = \( \sqrt{150^2 + 300^2} = \sqrt{22500 + 90000} = \sqrt{112500} \approx \mathbf{335.41} \text{ pixels}\)
Teacher Tip: Terminal vs. Initial
Remind students that subtraction is always (Terminal - Initial) . A common error is calculating (Initial - Terminal), which results in a vector pointing in the opposite direction.
Vector Algebra Slides Lesson 04
Vector Algebra
Scalar multiplication, algebraic addition, and linear combinations.
Adding Components
To add or subtract vectors algebraically, simply perform the operation on their corresponding components .
If \(\vec{u} = \langle a, b \rangle\) and \(\vec{v} = \langle c, d \rangle\):
\[ \vec{u} + \vec{v} = \langle a+c, \ b+d \rangle \]
It's that simple! No ruler or protractor required.
Example
\(\vec{u} = \langle 3, -1 \rangle\)
\(\vec{v} = \langle 4, 5 \rangle\)
\[ \vec{u} + \vec{v} = \langle 7, 4 \rangle \]
Scalar Multiplication
A **scalar** is a real number that "scales" a vector. It changes the magnitude and, if negative, reverses the direction.
\[ k\langle a, b \rangle = \langle ka, kb \rangle \]
This is like the distributive property for vectors!
v 2v (Double Length) -v (Reverse Dir)
Magnitude and Scalars
Multiplying a vector by a scalar \(k\) changes the magnitude by a factor of \(|k|\).
\[ ||k\vec{v}|| = |k| \cdot ||\vec{v}|| \]
Quick Check
If \(||\vec{v}|| = 10\):
\(||3\vec{v}||\) 30
\(||-2\vec{v}||\) 20
\(||0.5\vec{v}||\) 5
Linear Combinations
A **linear combination** of vectors \(\vec{u}\) and \(\vec{v}\) is any vector of the form:
\[ a\vec{u} + b\vec{v} \]
Example Problem
\(\vec{u} = \langle 1, 2 \rangle\)
\(\vec{v} = \langle -3, 0 \rangle\)
Find \(2\vec{u} - 3\vec{v}\)
Solution Step
\(2\langle 1, 2 \rangle - 3\langle -3, 0 \rangle\)
\(\langle 2, 4 \rangle + \langle 9, 0 \rangle\)
Result: \(\langle 11, 4 \rangle\)
Linear Combinations Worksheet Linear Combinations
Lesson 4: Vector Algebra & Scalar Operations
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\(\vec{u} = \langle 4, -2 \rangle\)
\(\vec{v} = \langle -1, 5 \rangle\)
\(\vec{w} = 3\mathbf{i} + 6\mathbf{j}\)
1
Basic Algebraic Operations
Perform the following operations. Write your final answer in component form.
A. \(\vec{u} + \vec{v}\)
Show work...
B. \(\vec{v} - \vec{u}\)
Show work...
C. \(4\vec{u}\)
Show work...
D. \(-3\vec{v}\)
Show work...
2
Linear Combinations
Calculate the resultant vector for each linear combination. Write final answers in \(\mathbf{i}, \mathbf{j}\) notation.
\[ 2\vec{u} + 3\vec{v} \]
Final Answer:
\[ \vec{w} - 2\vec{v} \]
Final Answer:
3
Bridge Support Challenge
A bridge support cable is represented by the vector \(\vec{C} = \langle 12, 16 \rangle\) (units in kN). Engineers need to modify the cable to handle double the vertical load without changing the horizontal support.
1. Find the new vector \(\vec{C}'\) representing the modified cable.
2. Calculate the magnitude of the original cable tension \(||\vec{C}||\) and the new cable tension \(||\vec{C}'||\).
3. Explain why this modification changes the angle (direction) of the cable.
Linear Combinations Answer Key Linear Combinations [Answer Key]
Lesson 4: Vector Algebra & Scalar Operations
Instructor Key
Ref: \(\vec{u} = \langle 4, -2 \rangle\), \(\vec{v} = \langle -1, 5 \rangle\), \(\vec{w} = \langle 3, 6 \rangle\)
1. Basic Algebraic Operations
A. \(\vec{u} + \vec{v}\)
\( \langle 4-1, -2+5 \rangle = \mathbf{\langle 3, 3 \rangle} \)
B. \(\vec{v} - \vec{u}\)
\( \langle -1-4, 5-(-2) \rangle = \mathbf{\langle -5, 7 \rangle} \)
C. \(4\vec{u}\)
\( 4\langle 4, -2 \rangle = \mathbf{\langle 16, -8 \rangle} \)
D. \(-3\vec{v}\)
\( -3\langle -1, 5 \rangle = \mathbf{\langle 3, -15 \rangle} \)
2. Linear Combinations
\[ 2\vec{u} + 3\vec{v} \]
\( 2\langle 4, -2 \rangle + 3\langle -1, 5 \rangle = \langle 8, -4 \rangle + \langle -3, 15 \rangle = \langle 5, 11 \rangle \)
Ans: \( 5\mathbf{i} + 11\mathbf{j} \)
\[ \vec{w} - 2\vec{v} \]
\( \langle 3, 6 \rangle - 2\langle -1, 5 \rangle = \langle 3, 6 \rangle - \langle -2, 10 \rangle = \langle 3-(-2), 6-10 \rangle = \langle 5, -4 \rangle \)
Ans: \( 5\mathbf{i} - 4\mathbf{j} \)
3. Bridge Support Challenge
1. Find the new vector \(\vec{C}'\)
Original: \(\langle 12, 16 \rangle\). Double vertical load means doubling the y-component. Horizontal remains 12. New y is \(16 \times 2 = 32\). New vector: \(\mathbf{\langle 12, 32 \rangle}\).
2. Calculate magnitudes
Original: \( ||\vec{C}|| = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = \mathbf{20} \text{ kN}\)
New: \( ||\vec{C}'|| = \sqrt{12^2 + 32^2} = \sqrt{144 + 1024} = \sqrt{1168} \approx \mathbf{34.18} \text{ kN}\)
3. Explain the angle change
The angle of the cable is determined by the ratio of the components (\(y/x\)). Since the vertical component increased while the horizontal stayed constant, the cable must become steeper (larger angle relative to horizontal) to handle the increased downward load.
Real World Vectors Slides Capstone Lesson
Real World Vectors
Modeling velocity in navigation and resolving forces in structural engineering.
APPLIED_PHYSICS_v1.0
Vector Navigation
In air navigation, the actual path of the plane (Ground Speed) is the **resultant** of the plane's motor (Airspeed) and the wind.
\[ \vec{V}_G = \vec{V}_A + \vec{V}_W \]
Airspeed (\(\vec{V}_A\)): Speed relative to the air.Wind Velocity (\(\vec{V}_W\)): Speed of air relative to ground.Ground Speed (\(\vec{V}_G\)): Actual speed/path.
Airspeed Vector Wind True Track (Ground Speed)
Static Force Equilibrium
For an object to remain stationary (in equilibrium), the sum of all force vectors acting on it must be the zero vector .
\[ \sum \vec{F} = \vec{0} \]
This means the sum of all x-components is 0 AND the sum of all y-components is 0.
Gravity (W) Tension 1 Tension 2
Inquiry: The Crosswind
"A pilot points her plane due North at an airspeed of 200 mph. A crosswind is blowing from the West at 40 mph. What is her true heading and speed?"
Step 1: Components
\[ \vec{V}_A = \langle 0, 200 \rangle \]
\[ \vec{V}_W = \langle 40, 0 \rangle \]
Step 2: Resultant
\[ \vec{V}_G = \langle 40, 200 \rangle \]
Magnitude: \( \sqrt{40^2 + 200^2} \approx 204 \text{ mph} \)
The Capstone Challenge
It's time to put your skills to the test. You will act as an Air Traffic Controller and a Structural Engineer to solve complex vector problems in the field.
Pick up your Navigation & Force Challenge Worksheet
Navigation & Force Worksheet Navigation & Force
Lesson 5: Applied Vector Case Studies
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Case Study 1: The Flight of the Osprey
The Osprey-X is flying at an airspeed of 500 km/h on a bearing due North. It encounters a strong tailwind blowing from the Southwest (\(45^\circ\) North of East) at 60 km/h.
1. Resolve the wind velocity \(\vec{V}_W\) into its North and East components.
2. Find the resultant velocity vector (Ground Speed) \(\vec{V}_G\) in component form.
3. Calculate the magnitude of the ground speed (how fast the plane is moving relative to the ground).
Case Study 2: Traffic Light Equilibrium
A traffic light weighing 300 Newtons is suspended by two cables. Cable 1 exerts a force \(\vec{T}_1 = \langle -100, 150 \rangle\). Cable 2 exerts a force \(\vec{T}_2 = \langle 100, 150 \rangle\).
1. Write the weight of the traffic light (\(\vec{W}\)) as a vector in component form.
2. Prove that the traffic light is in **static equilibrium** by finding the sum of all forces: \(\vec{T}_1 + \vec{T}_2 + \vec{W}\).
3. If Cable 1 snaps, describe the initial direction of the resultant force acting on the light.
Schematic Area
Expert Synthesis
"In your own words, explain how vectors bridge the gap between pure mathematics and physical reality. Use one example from this unit to support your answer."
Navigation & Force Answer Key Navigation & Force [Answer Key]
Lesson 5: Capstone Application Guide
Instructor Final Key
Case Study 1: The Flight of the Osprey
1. Resolve wind velocity \(\vec{V}_W\)
Wind from Southwest means pointing Northeast (\(45^\circ\)).
\(x = 60 \cdot \cos(45^\circ) \approx 42.43\)
\(y = 60 \cdot \sin(45^\circ) \approx 42.43\)
Ans: \(\vec{V}_W = \langle 42.43, 42.43 \rangle\)
2. Resultant velocity \(\vec{V}_G\)
Airspeed \(\vec{V}_A = \langle 0, 500 \rangle\) (Due North).
\( \vec{V}_G = \langle 0 + 42.43, 500 + 42.43 \rangle \)
Ans: \(\vec{V}_G = \langle 42.43, 542.43 \rangle\)
3. Magnitude of ground speed
\( ||\vec{V}_G|| = \sqrt{42.43^2 + 542.43^2} \approx \sqrt{1800 + 294230} \)
Ans: \(\approx 544.09 \text{ km/h}\)
Case Study 2: Traffic Light Equilibrium
1. Weight as a vector
Weight points straight down.
Ans: \(\vec{W} = \langle 0, -300 \rangle\)
2. Equilibrium Proof
Sum x: \(-100 + 100 + 0 = 0\)
Sum y: \(150 + 150 - 300 = 0\)
Since the sum of both components is 0, the object is in equilibrium.
3. Cable 1 Snaps
If \(\vec{T}_1\) is gone, the only remaining forces are \(\vec{T}_2 = \langle 100, 150 \rangle\) and \(\vec{W} = \langle 0, -300 \rangle\).
Net Force: \(\langle 100 + 0, 150 - 300 \rangle = \mathbf{\langle 100, -150 \rangle}\)
Direction: The light will swing Right and drop Down (Southeast direction).
Synthesis Grading Note
Students should recognize that vectors allow us to quantify intuitive physical concepts like "pushing" or "drifting." A strong answer would reference how trigonometry (geometry) and components (algebra) work together to predict precise physical outcomes, like an airplane's arrival time or a bridge's stability.
