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In physics, graphical vector representation provides a visual method for understanding the direction and magnitude of vectors. This is particularly useful in problems, such as the one involving the flight paths of an airplane, where comprehension of relative positions is necessary. Start by selecting a reference point, which remains constant throughout the representation. In the case of the plane, the base camp serves as this reference point. Stretch out a vector from this origin that represents the first journey to lake A. It must measure 280 km at an angle of 20° north of east. When drawing vectors, it's important to scale distances accurately for proper analysis. Once lake A is reached, proceed by drawing the next vector for the trip to Lake B. This vector should be 190 km at an angle of 30° west of north. Through this method, the final vector connecting lake B back to the base camp can be completed, visually outlining the path and showing the compounded vector components.

The resultant vector is the combined effect of two or more vectors. When two vectors are graphically combined, the resultant vector shows the final position or consequence after applying both vectors sequentially. In our exercise, the first vector starts at the base camp and travels 280 km to Lake A. The second vector starts from Lake A and travels 190 km to Lake B. To find the resultant vector from Lake B to the base camp:

• Draw both vectors tip-to-tail.
• Find the vector that directly connects Lake B to the base camp—a direct line starting at the end of the vector from Lake B and ending at the base camp.
• Measure this vector to determine the distance from Lake B back to the base camp.

This final vector provides both the direct distance and direction the plane must fly to return to the base camp from Lake B.

Trigonometric functions are essential tools in physics for solving vector-related problems. They help in breaking vectors into components, understanding angles and calculating distances. Consider the first vector from the problem:

• The length is 280 km, and its direction is 20° north of east.
• Components can be resolved using sine and cosine: the eastward component is found using cosine, and the northward component is found using sine.
• Similarly, for the second vector from Lake A to Lake B, components can be resolved at 30° west of north.

Using these trigonometric principles allows you to transform a vector into its constituents, making analysis of direction and subsequent calculations much more manageable.

Displacement in physics refers to the shortest distance from the starting point to the final position. In the given problem, calculating displacement involves using both the magnitude and direction of the resultant vector. Follow the steps below to accurately determine displacement:

• Resolve each journey's vector into components using trigonometric functions, like those detailed above.
• Add corresponding components: sum of all east/west and north/south components separately.
• Use the Pythagorean theorem to determine the magnitude of the resultant vector: \( \sqrt{x^2 + y^2} \).
• Calculate the angle using inverse tangent, where the angle \( \theta = \tan^{-1}(\frac{y}{x}) \).

The resulting value provides the direct distance, or displacement, and the angle gives the direction relative to your reference. This holistic calculation is crucial for effectively understanding motion in physical systems.