Vector addition and subtraction (triangle and parallelogram rules)

✖️ 1. Geometric addition: The tip-to-tail (triangle) rule and the parallelogram rule

🔺 Triangle Rule (Tip-to-Tail)

• Place the tail of the second vector at the tip of the first vector.
• The resultant vector goes from the tail of the first to the tip of the second.
• Order does not matter: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$.
• This is the most common method for adding vectors graphically.

Example: Walk 3 m east then 4 m north. The resultant is 5 m northeast.

💡 Think: Connect arrows head-to-tail like a chain.

▱ Parallelogram Rule

• Place both vectors tail-to-tail at the same starting point.
• Complete the parallelogram using the two vectors as adjacent sides.
• The diagonal from the common tail is the resultant.
• This gives the same answer as the triangle rule.

Example: Two forces of 3 N and 4 N at right angles form a diagonal of 5 N.

💡 Think: The diagonal cuts the parallelogram in half.

✖️ 2. Algebraic addition and subtraction using vector components

🧮 Component-Wise Addition

• Write each vector in component form: $\mathbf{u} = \langle u_x, u_y \rangle$ and $\mathbf{v} = \langle v_x, v_y \rangle$.
• Add corresponding components: $\mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle$.
• Subtraction works the same way: $\mathbf{u} - \mathbf{v} = \langle u_x - v_x, u_y - v_y \rangle$.
• This method is faster than drawing diagrams.
• Works in any dimension (2D, 3D, etc.).

Example: $\langle 3, 2 \rangle + \langle 1, 4 \rangle = \langle 4, 6 \rangle$.

💡 Think: Add x's together, add y's together.

✖️ 3. Geometric subtraction: $\mathbf{u} - \mathbf{v}$ as adding the opposite vector

↩️ Subtracting Vectors Geometrically

• Subtracting $\mathbf{v}$ is the same as adding $-\mathbf{v}$.
• The vector $-\mathbf{v}$ has the same length but opposite direction.
• Use tip-to-tail: place $-\mathbf{v}$ at the tip of $\mathbf{u}$.
• Alternatively: draw both vectors from the same point; $\mathbf{u} - \mathbf{v}$ points from the tip of $\mathbf{v}$ to the tip of $\mathbf{u}$.

Example: If $\mathbf{u} = \langle 5, 3 \rangle$ and $\mathbf{v} = \langle 2, 1 \rangle$, then $\mathbf{u} - \mathbf{v} = \langle 3, 2 \rangle$.

💡 Think: Flip the arrow, then add.

✖️ 4. Applications: Finding the net resultant force acting on an object or calculating relative velocity (e.g., a plane flying in a crosswind)

🛩️ Real-World Vector Addition

• Net force: Add all force vectors acting on an object to find the resultant.
• Relative velocity: Subtract velocities to find motion relative to another object.
• A plane's ground velocity = plane's air velocity + wind velocity.
• Use components for accuracy: break each vector into x and y parts.
• The magnitude of the resultant is found using the Pythagorean theorem.

Example: Plane flies 200 km/h east, wind blows 50 km/h north. Resultant speed is approximately 206 km/h northeast.

💡 Think: Combine all pushes and pulls into one.