add and subtract coplanar vectors

Scalars and Vectors: Adding & Subtracting Coplanar Vectors 🚀

1️⃣ What is a Vector?

A vector has both magnitude (how long) and direction (where it points). Think of a vector as an arrow: the length of the arrow tells you how big it is, and the arrowhead tells you which way it goes.

Example: The velocity of a car moving east at 60 km/h is a vector: $ \vec{v} = 60\,\text{km/h}\,\hat{\imath} $.

2️⃣ Scalars vs. Vectors

• Scalars: only magnitude (e.g., temperature 25 °C, mass 5 kg).
• Vectors: magnitude + direction (e.g., force, displacement).

3️⃣ Representing Coplanar Vectors in Components

For vectors lying in the same plane (xy‑plane), we write them as: $$ \vec{A} = (A_x, A_y) \quad \text{and} \quad \vec{B} = (B_x, B_y) $$ where $A_x$ and $B_x$ are the horizontal components, $A_y$ and $B_y$ are the vertical components.

4️⃣ Adding Coplanar Vectors

To add two vectors, add their corresponding components: $$ \vec{A} + \vec{B} = (A_x + B_x,\; A_y + B_y) $$

1. Write each vector in component form.
2. Sum the horizontal components: $C_x = A_x + B_x$.
3. Sum the vertical components: $C_y = A_y + B_y$.
4. Resulting vector: $ \vec{C} = (C_x, C_y) $.

5️⃣ Subtracting Coplanar Vectors

Subtraction is just adding the negative of the second vector: $$ \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) = (A_x - B_x,\; A_y - B_y) $$

1. Change the sign of each component of $\vec{B}$.
2. Proceed as in addition.

6️⃣ Example: Adding Two Vectors

Let $ \vec{A} = (3, 4) $ and $ \vec{B} = (1, -2) $.

Step 1: Add components: $C_x = 3 + 1 = 4$ $C_y = 4 + (-2) = 2$

Result: $ \vec{C} = (4, 2) $ The magnitude: $|\vec{C}| = \sqrt{4^2 + 2^2} = \sqrt{20} \approx 4.47$ units. Direction (angle from +x): $ \theta = \tan^{-1}\!\left(\frac{2}{4}\right) \approx 26.6^\circ $.

7️⃣ Example: Subtracting Two Vectors

Let $ \vec{A} = (5, 0) $ and $ \vec{B} = (2, 3) $.

Step 1: Subtract components: $D_x = 5 - 2 = 3$ $D_y = 0 - 3 = -3$

Result: $ \vec{D} = (3, -3) $ Magnitude: $|\vec{D}| = \sqrt{3^2 + (-3)^2} = \sqrt{18} \approx 4.24$ units. Direction: $ \theta = \tan^{-1}\!\left(\frac{-3}{3}\right) = -45^\circ $ (i.e., 315° from +x).

8️⃣ Quick Practice Problems

1. Find $ \vec{E} = (2, 5) + (-4, 1) $.
2. Find $ \vec{F} = (7, -3) - (2, 4) $.
3. Given $ \vec{G} = (0, 6) $ and $ \vec{H} = (3, 0) $, calculate $ \vec{G} + \vec{H} $ and its magnitude.

💡 Key Takeaways

• Vectors are arrows; scalars are just numbers.
• Component addition/subtraction is component‑wise.
• Always keep track of signs when subtracting.
• Use the Pythagorean theorem to find magnitudes.
• Angles can be found with $\tan^{-1}$ (inverse tangent).