91Ó°ÊÓ

In physics, vector addition is a crucial concept that helps us understand how different vector quantities combine. A vector is characterized by both magnitude and direction. When multiple electric fields act at a point due to different charges, we apply vector addition to determine the resulting electric field.

To add vectors, we each break them into components along common axes, such as the x and y axes. These components allow us to handle the vectors in a simpler, component-wise fashion. For example, in the given problem, we have vectors \(\vec{E}_{1}\) and \(\vec{E}_{2}\), where each vector represents an electric field at point P.

The vector sum (or resultant vector) is found by adding the x-components together and adding the y-components together separately. By recombining these resultant components, we obtain the complete resultant field vector. This step is foundational in fields ranging from electromagnetism to mechanics.

Breaking down vectors into their components is a vital method in vector operations, particularly for problems involving non-perpendicular directions.

Each vector can be split into perpendicular components: one along the x-axis and one along the y-axis.

• The x-component represents the vector's effect in the horizontal direction.
• The y-component shows the vertical influence.

For instance, the electric field \(\vec{E}_{1}\) from the exercise is purely in the vertical direction, meaning it has an x-component \(E_{1x} = 0\) and a y-component \(E_{1y} = 450 \, N/C\).

Conversely, \(\vec{E}_{2}\) points neither horizontally nor vertically only. It involves an angle, so trigonometric functions help us find its x and y components: \(E_{2x} = -600 \, N/C \, \cos(36.9^\circ)\) and \(E_{2y} = -600 \, N/C \, \sin(36.9^\circ)\).

Understanding components helps efficiently solve vector problems, enabling us to simplify and focus on a vector's individual directional properties.

After identifying vector components, calculating the magnitude and direction of the resultant vector is the final stage.

The magnitude, representing the vector's overall strength, can be derived using the Pythagorean theorem due to the right-angle relationship in the components:\[E = \sqrt{E_x^2 + E_y^2}\]Here, \(E_x\) and \(E_y\) are sums of the vector components identified in earlier steps.

Once we know the magnitude, we determine the direction or angle of the resultant vector. The angle is found using the inverse tangent function:\[\theta = \tan^{-1}\left(\frac{E_y}{E_x}\right)\]This gives an angle relative to the positive x-axis.

The exercise requires further adjustment since the resultant vector lies in a specified quadrant. In this case, we add \(180^\circ\) to correct the result, ensuring the direction matches the vector placement on the coordinate plane. By comprehensively finding these values, we obtain a precise description of the vector's impact.