1. Component Form: Write your vectors as $\langle x, y \rangle$.
2. Vector Addition: Add the X's and add the Y's. (Never mix them!).
3. Magnitude: Use $|\mathbf{v}| = \sqrt{x^2 + y^2}$.
4. Direction: Direction is the angle, not just the length.
Mark each quantity as either a Scalar (S) or a Vector (V).
The speed of a horse: 15 mph.
The wind: 20 mph from the East.
The temperature of the sun: 5,000 degrees.
The weight of a stone pulling downward: 10 Newtons.
Find the resultant vector $\mathbf{r} = \mathbf{a} + \mathbf{b}$.
$\mathbf{a} = \langle 5, 2 \rangle$ and $\mathbf{b} = \langle 3, 10 \rangle$
$\mathbf{a} = \langle -4, 6 \rangle$ and $\mathbf{b} = \langle 4, -2 \rangle$
The Rescue Mission: A helicopter flies $\langle 10, 0 \rangle$ miles East, then encounters a wind pushing it $\langle -2, -5 \rangle$. What is its actual position vector?
If vector $\mathbf{a} = \langle 3, 0 \rangle$ and $\mathbf{b} = \langle 0, 4 \rangle$... calculate $|v + v|$. Is it 7? Or is it 5? Why does the Pythagorean Theorem matter here?
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If $\mathbf{v} = \langle 3, -2 \rangle$... find $4\mathbf{v}$.
The Half-Zeal: If a missionary team starts with a plan $\mathbf{p} = \langle 10, 20 \rangle$, but their funding is cut in half ($k = 0.5$)... what is their new vector?
A vector is given by $\mathbf{v} = \langle 6, 8 \rangle$.
1. Calculate the Magnitude $|v|$.
2. Divide each component by the magnitude to find the Unit Vector $\mathbf{u}$.
3. Verify that the length of $\mathbf{u}$ is exactly 1.
Objective: Explain Vector Addition to a younger student using a toy.
The Activity:
1. Tie two strings to a toy car.
2. Have two children pull the strings in different directions.
3. Observe where the car goes.
The Lesson: "Our life is like that car. It goes where the 'Sum' of our pulls points. If we both pull toward God, the car moves twice as fast!"
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