Two airplanes leave an airport at the same time. The first flies 150 km / h in a direction of 32

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Two airplanes leave an airport at the same time. The first...

Two airplanes leave an airport at the same time. The first flies $150 \mathrm{km} / \mathrm{h}$ in a direction of $320^{\circ} .$ The second flies $200 \mathrm{km} / \mathrm{h}$ in a direction of $200^{\circ} .$ After 3 hr, how far apart are the planes?

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Key Concepts

Displacement Calculation

Displacement is determined by multiplying the speed by the time of travel. This process converts the given velocities into distances traveled by the objects over a period, laying the groundwork for subsequent vector analysis.

Vector Representation

Representing displacement as vectors involves specifying both magnitude and direction. This concept is pivotal for converting navigational directions, often given in degrees, into a format (typically rectangular coordinates) where vector addition or subtraction can be used to determine relative positions.

Polar to Rectangular Conversion

Converting polar coordinates (magnitude and angle) into rectangular coordinates using sine and cosine functions is essential for breaking down vectors into their horizontal and vertical components. This step simplifies the process of finding the difference between two displacement vectors by aligning them along common axes.

Law of Cosines

The Law of Cosines is used to compute the distance between two points when two sides of a triangle and the angle between them are known. This concept is especially useful in problems involving non-right triangles resulting from the vector paths of objects moving in different directions, allowing the calculation of the direct distance separating their endpoints.

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