Directed Journeys Slides Mathematics: Vectors
Directed Journeys
Exploring the Geometric Foundations of Vectors
The Flight Path Challenge
Imagine a plane is flying North at 500 mph.
Suddenly, a strong wind blows East at 80 mph.
Where does the plane actually end up relative to the ground?
Resultant Path?
Scalar vs. Vector
Scalar
A quantity described only by magnitude (size or amount).
Speed (55 mph)
Mass (10 kg)
Temperature (72°F)
Vector
A quantity described by both magnitude and direction .
Velocity (55 mph North)
Displacement (10m Up)
Force (15N East)
Directed Line Segments
Geometrically, we represent a vector as an arrow .
1
Length = Magnitude
How "much" of the quantity exists.
2
Arrowhead = Direction
Where the quantity is pointing.
3
Initial & Terminal Points
A vector starts at the "tail" and ends at the "tip".
Initial Point (Tail) Terminal Point (Tip) Magnitude
Vector Addition: Head-to-Tail
To find the Resultant of two vectors \(\vec{a}\) and \(\vec{b}\):
Step 1
Draw vector \(\vec{a}\) from its starting point.
Step 2
Draw 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 \(\vec{b}\) .
\(\vec{a}\) \(\vec{b}\) \(\vec{a} + \vec{b}\)
The Parallelogram Law
If two vectors \(\vec{u}\) and \(\vec{v}\) start from the same point :
Place tails of both vectors together.
Complete the parallelogram using dashed lines.
The diagonal from the common tail is the resultant \(\vec{u} + \vec{v}\).
Useful when multiple forces act on a single object!
Directed Journeys Worksheet Directed Journeys
Geometric Representation of Vectors
Name:
Date:
Head-to-Tail Method
Place the tail of the second vector at the tip of the first. The resultant starts at the first tail and ends at the second tip.
Parallelogram Rule
Place both tails at the same point. Form a parallelogram. The resultant is the diagonal starting from the common tail.
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Scalar vs. Vector
Identify each quantity as either a Scalar (S) or a Vector (V).
A car traveling 65 miles per hour.
A boat moving 15 knots due West.
The force of 50 Newtons pulling downward.
A temperature of 98.6 degrees Fahrenheit.
2
Vector Addition (Head-to-Tail)
Given the vectors \(\vec{u}\) and \(\vec{v}\) below, sketch the resultant vector \(\vec{w} = \vec{u} + \vec{v}\) using the head-to-tail method.
Given Vectors \(\vec{u}\) \(\vec{v}\)
Your Sketch: \(\vec{u} + \vec{v}\)
3
Parallelogram Rule
Use the parallelogram rule to find the resultant of vectors \(\vec{a}\) and \(\vec{b}\) that start at the same origin \(P\).
Given Vectors P \(\vec{a}\) \(\vec{b}\)
Your Sketch: Resultant
4
Application: The Plane and the Wind
A pilot points her aircraft due North with an airspeed of 400 km/h. A crosswind is blowing from West to East at 100 km/h.
1. Sketch the velocity vector of the plane (\(\vec{P}\)) and the velocity vector of the wind (\(\vec{W}\)) using the head-to-tail method.
2. Based on your sketch, will the plane's actual ground speed be 300 km/h, 400 km/h, 500 km/h, or more than 400 km/h? Explain your reasoning geometrically (without using formulas yet).
Vector Voyages Teacher Guide Teacher Facilitation Guide
Sequence: Vector Voyages
GR 11 MATH
Sequence Narrative
This sequence establishes vectors as foundational tools for both pure mathematics and physical science. We move from the conceptual (arrows and directions) to the analytical (components and dot products) to the applied (forces and mechanics). The core pedagogical shift is from 1D scalar math to 2D vector analysis where orientation matters as much as amount.
Lesson 1
Directed Journeys
Key Concept: Geometric Visualization
Focus on the distinction between scalars (distance) and vectors (displacement). Emphasize "tip-to-tail" logic. Common Pitfall: Students often try to add magnitudes directly (e.g., 3m + 4m = 7m) even when directions are perpendicular.
Lesson 2
Component Blueprint
Key Concept: Analytic Precision
Connecting vectors to the Cartesian plane. Introduction of \(\langle x, y \rangle\) and unit vectors \(i, j\). Magnitude is simply the Distance Formula/Pythagorean Theorem. Direction requires trig (\(\tan \theta = y/x\)).
Lesson 3
Vector Arithmetic
Key Concept: Linear Combinations
The beauty of components: we can add vectors by adding numbers. Explain scaling geometrically—negative scalars flip the arrow 180°. Use the "Tug of War" to introduce the concept of a zero resultant (equilibrium).
Lesson 4
Orthogonal Angles
Key Concept: Multi-dimensional Multiplication
Dot product as a scalar result. The critical test for 90° angles (orthogonality). Use the "Work" example to explain why the dot product is high when vectors align and zero when they are perpendicular.
Lesson 5
Mechanical Forces
Key Concept: Static Equilibrium
The synthesis of all skills. Breaking force vectors into components to solve systems of equations. Focus on the Traffic Light and Ramp problems as classic engineering scenarios.
Common Misconceptions
The Magnitude Sum: Students often calculate \(|\vec{a}+\vec{b}|\) as \(|\vec{a}| + |\vec{b}|\). Use the triangle inequality to show this is only true if they point in the exact same direction.
Tangent Ambiguity: When calculating \(\theta\), \(\arctan(y/x)\) only gives values in Quad I and IV. Students must manually adjust by adding 180° for vectors in Quad II and III.
The Dot Product Result: Students often expect the dot product to be another vector. Stress that it is a single number (scalar).
Discussion Prompts
"If you walk 3 miles North and 4 miles East, why is your displacement 5 miles but your total distance 7 miles?"
Component Blueprint Slides Mathematics: Vectors
Component Blueprint
Bridging Geometry and Algebra on the Cartesian Plane
Why Components?
Geometric sketches are great for intuition, but algebraic components give us precision.
Allows for exact calculations.
Simplifies complex additions.
Standard for physics & engineering.
v = <4, 4>
"How far right? How far up?"
Component Form: \(\langle v_x, v_y \rangle\)
\(\langle x, y \rangle\)
Ordered Pair Notation
\(xi + yj\)
Unit Vector Notation
\(i\) is the unit vector \(\langle 1, 0 \rangle\)
(1 unit along the x-axis)
\(j\) is the unit vector \(\langle 0, 1 \rangle\)
(1 unit along the y-axis)
Magnitude & Direction
If a vector \(\vec{v} = \langle a, b \rangle\):
Magnitude
The "Length" of the vector.
\(|\vec{v}| = \sqrt{a^2 + b^2}\)
Derived from the Pythagorean Theorem.
Direction Angle
The angle \(\theta\) from the positive x-axis.
\(\tan \theta = \frac{b}{a}\)
\( \theta = \arctan(\frac{b}{a}) \) (Check quadrant!)
Step-by-Step Conversion
Example: \(\vec{v} = \langle 3, -4 \rangle\)
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Plot it: Right 3 units, Down 4 units. (Quadrant IV)
2
Magnitude: \( \sqrt{3^2 + (-4)^2} = \sqrt{9+16} = 5 \)
3
Direction: \( \tan \theta = -4/3 \rightarrow \theta \approx -53.1^\circ \)
\(\theta\) \(\langle 3, -4 \rangle\)
Component Blueprint Worksheet Component Blueprint
Algebraic Representation & Magnitudes
11th Grade Mathematics
Unit: Vector Analysis
Student Name
Date
Part 1
Converting to Component Form
Write the component form \(\langle x, y \rangle\) and the unit vector form \(xi + yj\) for each vector shown on the grid below.
1 unit = 1 block
Component Form:
Unit Vector Form:
Component Form:
Unit Vector Form:
Part 2
Magnitude & Direction Calculations
For each vector, calculate the magnitude \(|\vec{v}|\) and the direction angle \(\theta\) (relative to the positive x-axis). Round answers to one decimal place.
| Vector Form | Magnitude \(|\vec{v}|\) (Show Work) | Direction \(\theta\) (Show Work) |
| --- | --- | --- |
| \(\langle 5, 12 \rangle\) | | |
| \(\langle -3, 6 \rangle\) | | |
| \(8i - 15j\) | | |
Part 3
Reverse Engineering
Draw the following vector on the grid provided, starting from the origin. Then, find the terminal point coordinates.
Vector: \(|\vec{u}| = 10, \theta = 150^\circ\)
Calculation for Terminal Point (\(x, y\)):
Final Terminal Point:
( ____ , ____ )
Vector Arithmetic Slides Mathematics: Vectors
Vector Arithmetic
Algebraic Operations & Geometric Scaling
Adding Vectors Algebraically
To add or subtract vectors in component form, simply add or subtract their corresponding components.
The Rule:
\(\vec{u} = \langle u_1, u_2 \rangle\)
\(\vec{v} = \langle v_1, v_2 \rangle\)
\(\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2 \rangle\)
Example:
\(\vec{a} = \langle 3, -2 \rangle\)
\(\vec{b} = \langle 1, 5 \rangle\)
\(\vec{a} + \vec{b} = \langle 3+1, -2+5 \rangle\)
= \(\langle 4, 3 \rangle\)
Scalar Multiplication
Algebraic Effect
Multiply each component by the scalar \(k\).
\(k \vec{v} = \langle k v_1, k v_2 \rangle\)
Geometric Effect
If \(k > 1\): Stretch (Longer)
If \(0 < k < 1\): Shrink (Shorter)
If \(k < 0\): Opposite Direction
\(\vec{v}\) \(2\vec{v}\) \(-1\vec{v}\)
Linear Combinations
Combining scalar multiplication and addition.
Problem:
Let \(\vec{u} = \langle 2, -1 \rangle\) and \(\vec{v} = \langle 3, 4 \rangle\).
Solution:
\(2\langle 2, -1 \rangle - 3\langle 3, 4 \rangle\)
\(\langle 4, -2 \rangle - \langle 9, 12 \rangle\)
= \(\langle -5, -14 \rangle\)
Hook: Tug of War
A crate is being pulled by two teams:
Team A: \(\vec{F}_A = \langle 10, 5 \rangle\) Newtons
Team B: \(\vec{F}_B = \langle -4, 12 \rangle\) Newtons
The Challenge:
What force vector \(\vec{F}_C\) must you apply so the crate does not move?
(Goal: \(\vec{F}_A + \vec{F}_B + \vec{F}_C = \langle 0, 0 \rangle\))
Vector Arithmetic Worksheet VECTOR ARITHMETIC
Algebraic Operations & Scaling
Mission File: 11-MATH-03
OPERATIVE:
01
Basic Operations
Perform the indicated operations for \(\vec{u} = \langle 4, -3 \rangle\) and \(\vec{v} = \langle -2, 6 \rangle\).
A) \(\vec{u} + \vec{v}\)
\(\langle \quad , \quad \rangle\)
B) \(\vec{v} - \vec{u}\)
\(\langle \quad , \quad \rangle\)
C) \(3\vec{u}\)
\(\langle \quad , \quad \rangle\)
D) \(-2\vec{v}\)
\(\langle \quad , \quad \rangle\)
02
Linear Combinations
Given \(\vec{a} = 2i + 5j\) and \(\vec{b} = -3i + j\), find the resultant vector \(\vec{r} = 4\vec{a} - 2\vec{b}\).
Final Result:
____________________
03
Geometric Interpretation
Given Vector \(\vec{w}\)
Sketch \(-0.5\vec{w}\)
Challenge: Static Equilibrium
Three ropes are pulling a central metal ring. Force \(\vec{F}_1 = \langle 15, -10 \rangle\) and force \(\vec{F}_2 = \langle -8, 22 \rangle\). Find the required force \(\vec{F}_3\) such that the sum of all forces is the zero vector \(\vec{0} = \langle 0, 0 \rangle\).
Show Algebraic Steps:
Final Force Vector \(\vec{F}_3\):
\(\langle \quad \quad , \quad \quad \rangle\)
Dot Product Slides Mathematics: Vectors
Orthogonal Angles
Mastering the Dot Product & Geometric Relationships
Why "Dot" Two Vectors?
In Physics, Work is done when a force moves an object.
"If you push a heavy box perfectly sideways, does it help lift the box up?"
No. Only the force in the direction of motion counts.
The Dot Product measures how much one vector "points along" another.
Force \(\perp\) Motion = No Work
The Algebraic Dot Product
The Formula
Multiply components and sum them up:
\(\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2\)
Important: The result is a SCALAR!
Example Calculation
\(\vec{u} = \langle 2, 4 \rangle\)
\(\vec{v} = \langle 3, -1 \rangle\)
\((2)(3) + (4)(-1)\)
\(6 + (-4)\)
\(= 2\)
Angle Between Vectors
The dot product is also related to the magnitudes and the cosine of the angle between them.
\(\cos \theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|}\)
Calculate the Dot Product first, then divide by the product of their lengths.
\(\theta\) \(\vec{u}\) \(\vec{v}\)
Orthogonal Vectors
The Condition: \(\vec{u} \cdot \vec{v} = 0\)
Two vectors are Orthogonal (perpendicular) if and only if their dot product is exactly zero.
Proof Sketch:
If \(\theta = 90^\circ\), then \(\cos(90^\circ) = 0\).
\(\langle 2, 3 \rangle \cdot \langle -3, 2 \rangle = 0\)
Orthogonal Angles Worksheet Orthogonal Angles
Dot Product & Orthogonality
Student Name:
Date:
1
The Scalar Dot Product
Calculate the dot product \(\vec{u} \cdot \vec{v}\) for each pair of vectors.
\(\vec{u} = \langle 3, 5 \rangle, \vec{v} = \langle 2, -1 \rangle\)
Show Calculation...
\(\vec{u} = \langle -4, 2 \rangle, \vec{v} = \langle 3, 6 \rangle\)
\(\vec{u} = 7i - 2j, \vec{v} = i + 3j\)
2
Angles Between Vectors
Find the angle \(\theta\) between vectors \(\vec{a} = \langle 4, 3 \rangle\) and \(\vec{b} = \langle -1, 2 \rangle\).
Step 1: Calculate \(\vec{a} \cdot \vec{b}\)
Step 2: Calculate \(|\vec{a}|\) and \(|\vec{b}|\)
Step 3: Solve for \(\theta\)
3
Testing for Orthogonality
Determine if the following pairs are Orthogonal:
\(\vec{u} = \langle 2, -5 \rangle, \vec{v} = \langle 10, 4 \rangle\)
Yes
No
\(\vec{a} = \langle 1, 1 \rangle, \vec{b} = \langle -1, 0 \rangle\)
Yes
No
Find the value of \(k\) that makes the vectors orthogonal:
\(\vec{u} = \langle 6, k \rangle, \vec{v} = \langle 2, 4 \rangle\)
Workspace...
Physics Connection: Work
A person pulls a sled 50 meters along flat ground (\(\vec{d} = \langle 50, 0 \rangle\)). They pull with a force of 100 Newtons at an angle of \(30^\circ\) (\(\vec{F} = \langle 100 \cos 30^\circ, 100 \sin 30^\circ \rangle\)).
Calculate the Work done (\(W = \vec{F} \cdot \vec{d}\)):
Mechanical Forces Slides Engineering & Mechanics
Mechanical Forces
Applying Vectors to Real-World Static Equilibrium and Velocity
Velocity in a Frame of Reference
When an object moves through a medium that is also moving (like a boat in a river or a plane in the wind), its Ground Velocity is the vector sum:
\(\vec{V}_{ground} = \vec{V}_{object} + \vec{V}_{medium}\)
River Crossing Problem
"If you point your boat straight across a river, but the current is pushing you downstream... where do you actually land?"
Static Equilibrium
The Rule
An object is in static equilibrium if it is at rest and the Sum of all Force Vectors acting on it is zero.
\(\sum \vec{F} = \vec{0}\)
Key Insight:
This means the sum of horizontal components is 0 AND the sum of vertical components is 0.
\(\sum F_x = 0\)
Left = Right
\(\sum F_y = 0\)
Up = Down
Case Study: Suspended Light
A 500 N traffic light is suspended by two cables. Cable 1 is at 30° and Cable 2 is at 45°.
Identify Forces: Tension 1, Tension 2, and Gravity.
Decompose: Find \((x, y)\) components for each.
Set up Equations:
\(T_{1x} + T_{2x} = 0\)
T1 T2 Gravity (W)
The Inclined Plane
Force Decomposition
When an object is on a ramp at angle \(\theta\), gravity is decomposed into two parts:
Parallel to ramp: \(mg \sin \theta\)
Perpendicular to ramp: \(mg \cos \theta\)
"This explains why a steeper ramp makes a ball roll faster—more gravity is pulling it parallel to the surface!"
Mechanical Forces Worksheet Mechanical Forces
Vector Applications & Equilibrium
Student Name:
Date:
1
The Crossing
A motorboat travels at a constant speed of 12 km/h relative to the water. The boat is pointed due East. The river has a current flowing due South at 5 km/h.
Tasks:
Find the magnitude of the boat's ground velocity.
Find the direction of the boat (the bearing from East).
Sketch Vectors Here
Ground Velocity Magnitude:
____________ km/h
2
The Traffic Light Challenge
A traffic light weighing 400 Newtons is supported by two cables. Cable A makes a 40° angle with the horizontal, and Cable B makes a 40° angle with the horizontal. (Note: The setup is symmetrical).
Step-by-Step Calculation (Decompose Force of Gravity and Cable Tensions):
Equation for Y-components:
Tension in Each Cable:
________ N
3
The Ramp
20°
A crate weighing 200 kg is sitting on a ramp inclined at 20°. Calculate the magnitude of the force component parallel to the ramp that is pulling the crate downward.
Show Formula and Calculation...
Critical Reflection:
Why is it that as the angle of a suspended traffic light cable gets closer to horizontal (0°), the tension in the cable approaches infinity? Use your vector equations to justify.
