Linear Combination Challenge Worksheet Linear Combination Challenge
Physics Application: Force Resultants
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Vector Reference Guide
Unit Vectors:
\( \mathbf{i} = \langle 1, 0 \rangle \)
\( \mathbf{j} = \langle 0, 1 \rangle \)
Conversion Formulas:
\( x = \|\mathbf{v}\| \cos \theta \)
\( y = \|\mathbf{v}\| \sin \theta \)
01
The Sled Tug-of-War
Two children are pulling a sled. Maya pulls with a force of 50 N at an angle of 30° . Leo pulls with a force of 40 N at an angle of 150° .
Step 1: Convert Maya's force (\( \mathbf{F}_M \)) to \( a\mathbf{i} + b\mathbf{j} \) form.
Step 2: Convert Leo's force (\( \mathbf{F}_L \)) to \( a\mathbf{i} + b\mathbf{j} \) form.
Step 3: Find the Resultant Force \( \mathbf{F}_R = \mathbf{F}_M + \mathbf{F}_L \) using linear combinations.
02
The Crosswind Navigation
An airplane is flying at an airspeed of 500 mph due North (90°). A strong crosswind is blowing at 50 mph from the Southwest (bearing 225°).
Step 1: Write Airplane velocity (\( \mathbf{v}_P \)) in \( a\mathbf{i} + b\mathbf{j} \).
Step 2: Write Wind velocity (\( \mathbf{v}_W \)) in \( a\mathbf{i} + b\mathbf{j} \).
Step 3: Calculate Groundspeed and Direction (Add \( \mathbf{v}_P \) and \( \mathbf{v}_W \) algebraically first).
03
The Hanging Sign (Statics)
A 100 N sign is supported by two cables. Cable 1 exerts a force \( \mathbf{T}_1 \) at 120°. Cable 2 exerts a force \( \mathbf{T}_2 \) at 60°. The downward force of gravity is \( \mathbf{F}_G = -100\mathbf{j} \). If the sign is in equilibrium, then \( \mathbf{T}_1 + \mathbf{T}_2 + \mathbf{F}_G = 0 \).
Express the horizontal (\( \mathbf{i} \)) and vertical (\( \mathbf{j} \)) components as a system of equations to find \( \|\mathbf{T}_1\| \) and \( \|\mathbf{T}_2\| \).
Force Fields Presentation Force Fields
Mastering Linear Combinations of \( \mathbf{i} \) and \( \mathbf{j} \)
Warm-up: What's a "Unit"?
Unit Circle
What makes it a "Unit" circle?
Unit Price
What does "Price per Unit" tell us?
"In math and physics, a 'unit' usually implies a magnitude of exactly 1."
The Standard Basis
Watch for the definitions of \( \mathbf{i} \) and \( \mathbf{j} \).
Embedded media
Key Focus: How do we turn \( \langle x, y \rangle \) into \( x\mathbf{i} + y\mathbf{j} \)?
The Building Blocks
i
\( \langle 1, 0 \rangle \)
Horizontal Unit Vector
j
\( \langle 0, 1 \rangle \)
Vertical Unit Vector
Vector \( \mathbf{v} = \langle a, b \rangle \) is just a linear combination:
3i
4j
Why use \( \mathbf{i} \) and \( \mathbf{j} \)?
Algebraic Ease
Adding vectors becomes simple polynomial addition. Combine like terms!
Calculus Prep
Standard notation for derivatives of vector-valued functions.
Physics Logic
Isolates horizontal work from vertical gravity instantly.
RAPID FIRE!
Grab your whiteboards. I show component form, you write linear combination form.
Example: \( \langle 3, -2 \rangle \rightarrow 3\mathbf{i} - 2\mathbf{j} \)
Prompt 1
\( \langle 5, 8 \rangle \)
5i + 8j
Prompt 2
\( \langle -4, 0 \rangle \)
-4i
Prompt 3
\( \langle 12, -7 \rangle \)
12i - 7j
Challenge Time
Refer to your "Linear Combination Challenge" Worksheet.
Mission 1: Sled
Break down two tension forces. Find the total force helping the sled move.
Mission 2: Crosswind
Combine airplane thrust and wind resistance. Where is the plane actually going?
Final Verdict
On your whiteboard, write the Resultant Vector for Mission 1 in \( a\mathbf{i} + b\mathbf{j} \) form.
Hint
Sum the \( \mathbf{i} \)'s.
Sum the \( \mathbf{j} \)'s.
Vector Victory Answer Key Answer Key & Teacher Guide
Linear Combination Challenge
Precalculus / Physics
Lesson Facilitation
Warm-up (5 min): Prompt students to think about a "unit" as a basis of measurement. In the unit circle, the "unit" is 1 unit of distance from origin. In unit vectors, \( \mathbf{i} \) is 1 unit of distance in the x-direction.
Video (10 min): Focus on the shift from \( \langle x, y \rangle \) to \( x\mathbf{i} + y\mathbf{j} \). Ask: "Why is \( 3\mathbf{i} + 4\mathbf{j} \) more mathematically useful than the bracket notation?"
Main Activity (20 min): Students often struggle with \( \|\mathbf{v}\| \cos \theta \) when angles are in different quadrants. Encourage them to draw the reference triangle.
01: The Sled Tug-of-War
Maya's Force (\( \mathbf{F}_M \)):
\( 50 \cos 30^\circ \mathbf{i} + 50 \sin 30^\circ \mathbf{j} \)
\( 25\sqrt{3}\mathbf{i} + 25\mathbf{j} \approx 43.3\mathbf{i} + 25\mathbf{j} \)
Leo's Force (\( \mathbf{F}_L \)):
\( 40 \cos 150^\circ \mathbf{i} + 40 \sin 150^\circ \mathbf{j} \)
\( -20\sqrt{3}\mathbf{i} + 20\mathbf{j} \approx -34.6\mathbf{i} + 20\mathbf{j} \)
Resultant Force (\( \mathbf{F}_R \)):
\( (43.3 - 34.6)\mathbf{i} + (25 + 20)\mathbf{j} = \mathbf{8.7\mathbf{i} + 45\mathbf{j}} \)
Magnitude: \( \approx 45.8 \text{ N} \). Direction: \( \approx 79.1^\circ \)
02: The Crosswind Navigation
Plane Velocity (\( \mathbf{v}_P \)):
\( 500 \cos 90^\circ \mathbf{i} + 500 \sin 90^\circ \mathbf{j} \)
\( 0\mathbf{i} + 500\mathbf{j} \)
Wind Velocity (\( \mathbf{v}_W \)):
\( 50 \cos 225^\circ \mathbf{i} + 50 \sin 225^\circ \mathbf{j} \)
\( -25\sqrt{2}\mathbf{i} - 25\sqrt{2}\mathbf{j} \approx -35.4\mathbf{i} - 35.4\mathbf{j} \)
Groundspeed Velocity:
\( \mathbf{-35.4\mathbf{i} + 464.6\mathbf{j}} \)
Groundspeed: \( \approx 466 \text{ mph} \). Direction: \( \approx 94.4^\circ \)
03: The Hanging Sign (Challenge)
Horizontal Equation (Sum of \( \mathbf{i} \)): \( \|\mathbf{T}_1\| \cos 120^\circ + \|\mathbf{T}_2\| \cos 60^\circ = 0 \)
Since \( \cos 120^\circ = -1/2 \) and \( \cos 60^\circ = 1/2 \), this implies \( \|\mathbf{T}_1\| = \|\mathbf{T}_2\| \).
Vertical Equation (Sum of \( \mathbf{j} \)): \( \|\mathbf{T}_1\| \sin 120^\circ + \|\mathbf{T}_2\| \sin 60^\circ - 100 = 0 \)
Substituting: \( \|\mathbf{T}_1\| \frac{\sqrt{3}}{2} + \|\mathbf{T}_1\| \frac{\sqrt{3}}{2} = 100 \rightarrow \sqrt{3} \|\mathbf{T}_1\| = 100 \)
