Resolution of a Vector
Why Do We Need to Resolve Vectors?
Imagine you are pushing a lawnmower. You don't push it straight down into the ground; you push it at an angle, forward and slightly downward. Your push is a single force, a vector. However, its effect can be thought of in two separate ways: one part of your push moves the lawnmower forward, and another part presses it into the ground. The forward part is useful for the task, while the downward part is not only wasted but also makes the mower harder to push. To analyze this situation mathematically, we break down, or resolve, your single pushing force into two imaginary components: one horizontal (causing forward motion) and one vertical (pushing into the ground).
This is the essence of vector resolution. It allows us to:
- Simplify Problems: It is much easier to add and subtract vectors that lie along the same straight line. By converting diagonal vectors into perpendicular (right-angled) components, we can work with simple one-dimensional math along each axis separately.
- Understand Effects: We can isolate and study the effect of a vector in a specific direction. For example, in projectile motion, we resolve the initial velocity into horizontal and vertical components to independently analyze the constant horizontal motion and the vertically accelerated motion.
- Calculate Net Effects: When multiple forces act on an object at different angles, resolving them into components along common axes (like x and y) is the only practical way to find the overall, or net, force.
The Mathematics of Resolution: Trigonometry to the Rescue
The most common type of resolution is into two perpendicular components, usually along the horizontal (x) and vertical (y) axes. The mathematical tool we use for this is trigonometry, specifically the sine and cosine functions.
Consider a vector $\vec{R}$ with magnitude $R$ making an angle $\theta$ with the positive x-axis. We want to find its two components: $\vec{R_x}$ along the x-axis and $\vec{R_y}$ along the y-axis.
The x-component (horizontal): $R_x = R \cos\theta$
The y-component (vertical): $R_y = R \sin\theta$
These formulas come directly from the definitions of cosine and sine in a right-angled triangle. The original vector $\vec{R}$ is the hypotenuse, the x-component $\vec{R_x}$ is the adjacent side to the angle $\theta$, and the y-component $\vec{R_y}$ is the opposite side.
It is crucial to pay attention to the angle used. If the angle is given with respect to the y-axis, the sine and cosine roles would be reversed. A good practice is to always identify which sides of the triangle are adjacent and opposite relative to the given angle.
| Quadrant | X-Component ($R_x = R\cos\theta$) | Y-Component ($R_y = R\sin\theta$) |
|---|---|---|
| I (0° to 90°) | Positive | Positive |
| II (90° to 180°) | Negative | Positive |
| III (180° to 270°) | Negative | Negative |
| IV (270° to 360°) | Positive | Negative |
Step-by-Step Guide to Resolving a Vector
Let's break down the process into a simple, repeatable procedure.
- Identify the Vector and its Properties: Note the magnitude ($R$) of the vector and the direction angle ($\theta$). Always be clear about what reference line the angle is measured from (usually the positive x-axis).
- Draw a Sketch: Draw the vector on a coordinate plane. Sketch a right-angled triangle by dropping perpendiculars from the tip of the vector to the x and y axes. The vector itself is the hypotenuse.
- Apply Trigonometric Functions:
- To find the component adjacent to the angle $\theta$, use the cosine function.
- To find the component opposite to the angle $\theta$, use the sine function.
- Determine the Signs: Based on the quadrant in which the vector lies (see table above), assign the correct positive or negative sign to each component.
- State the Result: Express the original vector in terms of its components. This is often written as $\vec{R} = (R_x, R_y)$ or $\vec{R} = R_x\hat{i} + R_y\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors along the x and y axes, respectively.
Practical Application: Analyzing Forces on an Inclined Plane
One of the most classic applications of vector resolution is analyzing the motion of an object on an inclined plane, like a block sliding down a ramp. The force of gravity acts straight down, but its effect on the block is not straight down the ramp. We resolve the weight vector ($\vec{W} = mg$) into two components:
- Component parallel to the incline ($W_{\parallel}$): This component is responsible for pulling the block down the ramp. It is calculated as $W_{\parallel} = mg \sin\theta$, where $\theta$ is the angle of the incline.
- Component perpendicular to the incline ($W_{\perp}$): This component presses the block against the surface of the ramp. It is calculated as $W_{\perp} = mg \cos\theta$.
By resolving the weight, we can now understand that the force causing acceleration down the ramp is $mg \sin\theta$, not the full weight $mg$. This explains why objects accelerate more slowly on a gentle slope (small $\theta$, small $\sin\theta$) and faster on a steep slope (large $\theta$, large $\sin\theta$).
Common Mistakes and Important Questions
Q: I always get confused about when to use sine and when to use cosine. Is there a simple trick?
Q: Can a vector be resolved into components that are not at right angles?
Q: What is the difference between a resultant vector and a component vector?
Footnote
1. Vector: A quantity that has both magnitude (size) and direction. Examples include force, velocity, and displacement.
2. Component: One of the parts into which a vector has been resolved. These components are themselves vectors.
3. Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
4. Unit Vector ($\hat{i}$, $\hat{j}$): A vector with a magnitude of exactly 1, used to specify direction. $\hat{i}$ points in the positive x-direction, and $\hat{j}$ points in the positive y-direction.