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} = (Ax, Ay) \quad \text{and} \quad \vec{B} = (Bx, By) \$

where \$Ax\$ and \$Bx\$ are the horizontal components, \$Ay\$ and \$By\$ are the vertical components.

4️⃣ Adding Coplanar Vectors

To add two vectors, add their corresponding components:

\$ \vec{A} + \vec{B} = (Ax + Bx,\; Ay + By) \$

1. Write each vector in component form.
2. Sum the horizontal components: \$Cx = Ax + B_x\$.
3. Sum the vertical components: \$Cy = Ay + B_y\$.
4. Resulting vector: \$ \vec{C} = (Cx, Cy) \$.

5️⃣ Subtracting Coplanar Vectors

Subtraction is just adding the negative of the second vector:

\$ \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) = (Ax - Bx,\; Ay - By) \$

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).