Vector Basics: quick revision, 16 bites

Vectors vs. Scalars

Magnitude alone is not enough.

In physics, quantities are either **scalars** or **vectors**. A scalar has only magnitude (size), like temperature or mass. A vector has both magnitude and direction, like velocity or force. This distinction is crucial because many physical laws require direction to make accurate predictions.

Takeaway. Vectors include direction; scalars do not.

What is a Vector?

Arrow with purpose.

A **vector** is a mathematical object with **magnitude** (length) and **direction** (angle). It is often represented as an arrow: the length shows magnitude, the arrowhead shows direction. Common notations include $\vec{v}$ or $\mathbf{v}$.

Takeaway. A vector has both size and direction, shown as an arrow.

Vector Components

Breaking vectors into parts.

A vector can be split into **components** along perpendicular axes (usually x and y). For a vector $\vec{v}$ with magnitude $v$ and angle $\theta$ from the x-axis:

Takeaway. Components make vector math easier.

Magnitude from Components

Pythagoras helps vectors.

If you know the components $(v_x, v_y)$, you can find the magnitude using the Pythagorean theorem:

Takeaway. Magnitude is the square root of sum of squares of components.

Direction from Components

Angle from components.

The direction angle $\theta$ of a vector can be found using the inverse tangent of the ratio of components:

Takeaway. Use arctan to find direction, but check the quadrant.

Adding Vectors Graphically

Tip-to-tail method.

Takeaway. Graphical addition: tip-to-tail gives the resultant.

Adding Vectors by Components

Add x and y separately.

To add vectors $\vec{a} = (3, 4)$ and $\vec{b} = (1, -2)$:

Takeaway. Add components, then compute magnitude and direction.

Forgetting Direction in Addition

Direction matters in sum.

A common mistake is adding vectors like scalars, ignoring direction. For example, walking 5 m East then 5 m West results in zero displacement, not 10 m. Always consider direction when adding vectors.

Takeaway. Vector addition requires direction, not just magnitude.

Vectors in Navigation

Pilots use vectors daily.

Aircraft and ships use vectors for navigation. The plane's velocity relative to air, wind velocity, and ground velocity are all vectors. Adding them gives the actual path over ground. This is why pilots adjust heading to compensate for wind.

Takeaway. Vectors help calculate actual motion in wind or currents.

Force Vectors in Structures

Bridges balance forces.

In engineering, forces acting on a bridge are vectors. Each support experiences forces at specific angles. Engineers add these vectors to ensure the net force is zero (equilibrium), keeping the structure stable. Failure to account for vector directions can lead to collapse.

Takeaway. Vector addition ensures structural stability.

Remembering Components

Cos for x, sin for y.

To remember which component uses cosine: think **C** for **C**lose (adjacent to angle) and **S** for **S**tand (opposite). So $v_x = v \cos \theta$ (adjacent) and $v_y = v \sin \theta$ (opposite).

Takeaway. Cos for adjacent (x), sin for opposite (y).

Check Quadrant for Angle

Arctan can mislead.

When finding direction from components, the inverse tangent gives an angle in the first or fourth quadrant. If both components are negative, the vector points to the third quadrant. Add 180° to the arctan result to get the correct angle. Always sketch the vector to verify.

Takeaway. Always adjust arctan based on component signs.

Vectors in Computer Graphics

Pixels move with vectors.

In video games and animation, vectors are used to move characters, simulate physics, and render lighting. Every 3D object's position, rotation, and velocity are stored as vectors. Without vectors, modern graphics would not exist.

Takeaway. Vectors power computer graphics and game physics.

Vectors as Treasure Map

Follow the arrow to gold.

Think of a vector as a treasure map instruction: "Walk 10 paces East" (magnitude 10, direction East). If you only know "10 paces" without direction, you'd be lost. Vectors give both distance and heading, just like a good map.

Takeaway. Vectors are like map directions: magnitude + direction.

Always Decompose First

Break before you add.

When adding multiple vectors, always decompose each into x and y components first. Then sum all x-components and all y-components separately. Finally, combine the totals to get the resultant vector. This method avoids errors from graphical approximations.

Takeaway. Decompose, sum components, then recompose.

Net Force as Vector Sum

All forces add as vectors.

Newton's second law $\vec{F}_{net} = m\vec{a}$ requires vector addition of forces. If multiple forces act on an object, the net force is the vector sum. This determines the object's acceleration direction and magnitude. Ignoring vector nature leads to wrong predictions.

Takeaway. Net force is the vector sum of all forces.