# Resolving Vectors ## What is Resolving? - **Combining vectors** works in reverse: two vectors can be represented by a single resultant vector - **Resolving a vector** is the opposite process of adding vectors - A single resultant vector can be broken down and represented by two vectors - These two vectors, when combined, have the same effect as the original one

**Components** are the parts of a vector when it is broken down into two perpendicular directions

## Resolving into Horizontal and Vertical Components - Consider a force vector of magnitude $F$ at an angle $\theta$ to the horizontal - This vector can be resolved into its **horizontal** and **vertical** components using trigonometry ### Component Equations - **Horizontal component:** $F_x = F \cos \theta$ - **Vertical component:** $F_y = F \sin \theta$

The horizontal component uses cosine, the vertical component uses sine

### How Components Vary with Angle The chart below shows how the horizontal and vertical components change as the angle increases from 0° to 90° for a vector with magnitude 100 N:

Key observations: - At 0°: the vector is entirely horizontal ($F_x = F$, $F_y = 0$) - At 90°: the vector is entirely vertical ($F_x = 0$, $F_y = F$) - At 45°: both components are equal ($F_x = F_y = 0.707F$) - As one component increases, the other decreases ### How to Remember Which is Which - The **cosine** component is always adjacent to the angle - Think of making a "cos sandwich" - the angle $\theta$ is sandwiched between the main vector $F$ and the horizontal component $F \cos \theta$ - If you imagine a right-angled triangle with the vector as the hypotenuse, $\cos \theta$ gives you the side touching (adjacent to) the angle ## Example: Forces on an Inclined Plane - Objects on slopes (inclined planes) are a common scenario where resolving vectors is essential - An **inclined plane** is a flat surface tilted at an angle $\theta$ ### Choosing the Right Axes - Instead of horizontal and vertical, it's more useful to think **parallel** and **perpendicular** to the slope - This choice makes the mathematics much simpler ### Weight Components on a Slope - The **weight** $W$ acts vertically downwards - The **normal reaction force** $R$ acts perpendicular to the slope (pushing up from the surface) - The weight vector can be split into: - **$W \cos(\theta)$** perpendicular to the slope - **$W \sin(\theta)$** parallel to the slope (down the slope)

If there's no friction, $W \sin(\theta)$ causes the object to accelerate down the slope

### Equilibrium Perpendicular to the Slope - The object doesn't move perpendicular to the slope - Therefore, the forces perpendicular to the slope must be balanced - This means: $R = W \cos(\theta)$ ## Worked Example: Aircraft Thrust A jet engine provides a thrust of 180 kN when the aircraft is climbing at 22° from the horizontal. Calculate the horizontal and vertical components of the thrust force. ### Answer: **Step 1: Sketch a vector triangle** Draw the thrust force at 22° from the horizontal, with the angle clearly marked. Identify which components you need (horizontal and vertical). **Step 2: Calculate the horizontal component** $$\text{Horizontal} = 180 \times \cos(22°) = 167 \text{ kN}$$ **Step 3: Calculate the vertical component** $$\text{Vertical} = 180 \times \sin(22°) = 67.4 \text{ kN}$$ ## Key Points Summary - Resolving is the **inverse operation** of combining vectors - A resultant vector can be split into **two perpendicular components** - Use **trigonometry** to find component magnitudes: - Adjacent component (usually horizontal): $F \cos \theta$ - Opposite component (usually vertical): $F \sin \theta$ - For **inclined planes**, resolve parallel and perpendicular to the slope - The component parallel to a slope is $W \sin(\theta)$ - The component perpendicular to a slope is $W \cos(\theta)$

Always draw a clear vector diagram first - it helps you identify which trigonometric function to use

## Common Applications | Scenario | Components to Resolve | |----------|----------------------| | Projectile motion | Horizontal and vertical velocity | | Forces on slopes | Parallel and perpendicular to slope | | Tension in cables at angles | Horizontal and vertical components of tension | | Aircraft or rocket thrust | Horizontal (forward) and vertical (lift) |