Question
A displacement vector, has magnitude \( d=197 \mathrm{~m} \) and points in a direction \( \theta=21.6^{\circ} \) measured counterclockwise from the positive \( y \) axis Enter an expression for the horizontal component of the vector d using the symbols provided.
Ask by Ruiz Barker. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The horizontal component of the vector is \( d_x = 197 \, \text{m} \times \sin(21.6^\circ) \).
Solution
The horizontal component (\( d_x \)) of the displacement vector can be expressed using the magnitude \( d \) and the angle \( \theta \) as follows:
\[
d_x = d \cdot \sin(\theta)
\]
**Explanation:**
- The angle \( \theta = 21.6^\circ \) is measured counterclockwise from the positive \( y \)-axis.
- To find the horizontal ( \( x \)-axis) component, you use the sine of the angle:
\[
d_x = d \sin(\theta)
\]
So, plugging in the values:
\[
d_x = 197 \, \text{m} \cdot \sin(21.6^\circ)
\]
This expression gives you the horizontal component of the displacement vector.
**Final Answer:**
\[
d_x = d \times \sin{\theta}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
The horizontal component of the displacement vector can be calculated using trigonometric functions. Since the angle \( \theta \) is measured from the positive \( y \) axis, the horizontal component \( d_x \) can be expressed as \( d_x = d \cdot \sin(\theta) \). Substituting the given values, we have: \( d_x = 197 \cdot \sin(21.6^{\circ}) \). So, when you plug in the values, you'll get the precise horizontal component of the vector. Enjoy calculating! And if you're curious about the vertical component too, it can be calculated using the cosine function, giving you the full picture of your vector's breakdown! Keep exploring those angles and magnitudes!
