If a vector $a$ makes an angle $\theta$ with the x-axis, what is the magnitude of its x-component $a_x$?

• A. $a\cdot \sin \left(\theta \right)$
• B. $a\cdot \tan \left(\theta \right)$
• C. $a\cdot \cos \left(\theta \right)$
• D. $a$

Answer: (C)

Verified step by step guidance

1. Identify the vector \( \mathbf{a} \) and the angle \( \theta \) it makes with the x-axis. The vector \( \mathbf{a} \) has a magnitude \( a \) and direction given by \( \theta \).
2. Recall that the x-component of a vector is the projection of the vector onto the x-axis. This can be found using trigonometry, specifically the cosine of the angle between the vector and the x-axis.
3. Use the formula for the x-component of the vector: \( a_x = a \times \cos(\theta) \). This comes from the definition of cosine in a right triangle, where the adjacent side corresponds to the x-component.
4. Understand why sine or tangent are not correct here: sine corresponds to the y-component (perpendicular to x-axis), and tangent is the ratio of y-component to x-component, not a component itself.
5. Thus, the magnitude of the x-component of vector \( \mathbf{a} \) is given by \( a_x = a \cdot \cos(\theta) \).