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Understanding trigonometry is crucial when dealing with vector components, especially for vectors that make angles with the Cartesian axes. Vectors like \(\vec{r}\) can be resolved into their horizontal and vertical components using trigonometric functions such as sine and cosine.
- The x-component of a vector, \(r_x\), is found by multiplying the vector's magnitude \(r\) by the cosine of the angle it makes with the x-axis, \(\theta\). Thus: \(r_x = r\cos(\theta)\). - Similarly, the y-component, \(r_y\), is obtained by multiplying the magnitude by the sine of the angle: \(r_y = r\sin(\theta)\).
In the given problem, the vector \(\vec{r}\) points in the northwesterly direction, forming a \(30^{\circ}\) angle with the x-axis. To accurately derive the components, apply these trigonometric relationships considering the orientation. Here, the x-component is negative due to the vector pointing west (negative x-direction).

Graphical verification offers a visual confirmation of the calculations involved in resolving a vector's components. This method is particularly beneficial for understanding and confirming analytical solutions through a tangible approach.
To graphically verify the vector components: - Draw the vector \(\vec{r}\) on graph paper using an appropriate scale. - Start from the origin and ensure it forms a \(30^{\circ}\) angle with the x-axis pointing northwest. - Construct the right-angled triangle by dropping perpendicular lines from the vector to the x and y axes, representing the vector components.
Measure the lengths of these lines. If the diagram is accurate, these measurements should closely match the computed values for \(r_x\) and \(r_y\). This exercise not only reinforces understanding but also encourages the practical application of theoretical concepts.

Direction angles are pivotal in determining a vector's orientation in a coordinate system. The angle here refers to the one between the vector itself and a reference axis, often the x-axis, measured in a counterclockwise direction.
In this scenario, the vector \(\vec{r}\) makes a \(30^{\circ}\) angle with the x-axis. This particular case involves a vector directed in the northwest direction, meaning that: - The angle is measured with respect to the negative x-axis. - The direction angles allow us to define and calculate the vector's components accurately by considering the appropriate trigonometric function for each component.
By understanding the concept of direction angles, one can effectively describe a vector's orientation in any plane, paving the way for precise vector manipulations and applications in physics and engineering.