represent a vector as two perpendicular components
Scalars and Vectors
What is a scalar?
Scalars are quantities that only have magnitude. They don’t point in any direction. Examples: speed (50 km h⁻¹), mass (10 kg), temperature (30 °C).
What is a vector?
Vectors have both magnitude and direction. They are usually shown as arrows. Example: a velocity of 30 m s⁻¹ to the east. The arrow points east and its length represents 30 m s⁻¹.
Representing a vector with perpendicular components
Any vector can be split into two perpendicular components along the x‑axis and y‑axis. Think of a skateboarder moving diagonally across a field – the horizontal and vertical parts of his motion are the components.
Mathematically we write:
$$\vec{v} = v_x\,\hat{i} + v_y\,\hat{j}$$where $v_x$ is the horizontal component and $v_y$ is the vertical component.
| Component | Symbol | Unit |
|---|---|---|
| Horizontal | $v_x$ | m s⁻¹ |
| Vertical | $v_y$ | m s⁻¹ |
Example: Finding components of a 10 m s⁻¹ velocity at 30° above the horizontal
- Identify the magnitude $v = 10$ m s⁻¹ and the angle $\theta = 30^\circ$.
- Compute the horizontal component: $v_x = v \cos\theta$.
- Compute the vertical component: $v_y = v \sin\theta$.
- Write the vector in component form.
Calculations:
$$v_x = 10 \cos 30^\circ = 8.66 \text{ m s}^{-1}$$ $$v_y = 10 \sin 30^\circ = 5.00 \text{ m s}^{-1}$$Thus the vector is:
$$\vec{v} = 8.66\,\hat{i} + 5.00\,\hat{j}\ \text{m s}^{-1}$$Exam Tip 🚀
When the exam asks for components:
- Use $v_x = v \cos\theta$ and $v_y = v \sin\theta$ if the angle is measured from the x‑axis.
- Check the quadrant of the angle to decide the sign of each component.
- If components are given, you can find the magnitude with $v = \sqrt{v_x^2 + v_y^2}$ and the direction with $\theta = \tan^{-1}(v_y/v_x)$.
- Always round to the correct number of significant figures as specified in the question.
Common Mistakes to Avoid
- Mixing up $\cos$ and $\sin$ for the wrong component.
- Forgetting to convert degrees to radians if the calculator is set to radian mode.
- Using the wrong sign for components in the second or third quadrants.
- Not checking significant figures or units in the final answer.
Remember: vectors are like arrows on a map – the length tells you how big, and the direction tells you where. Splitting them into perpendicular components is just like breaking a pizza slice into a horizontal and a vertical slice – each part is easier to handle, and together they give you the whole picture. 🍕📐
Revision
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