Chapter 1: Vectors — General Physics PHYS1251

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In physics, we deal with two types of quantities: scalars, which have only magnitude, and vectors, which have both magnitude and direction. Temperature is a scalar, but velocity is a vector because it tells you how fast and which way you are moving. Vectors are represented as arrows. The length of the arrow shows the magnitude, and the arrowhead shows the direction. To add two vectors geometrically, place the tail of the second vector at the head of the first, and draw the resultant from the first tail to the final head. Any vector can be broken into components along the x and y axes. The x-component equals the magnitude times cosine of the angle, and the y-component equals the magnitude times sine of the angle. Together, these two components fully describe the vector using a right triangle. In Sample Problem 1, a hiker uses three displacements to get as far as possible from base camp. By choosing vectors a, b, and negative c in a head-to-tail arrangement, the total x-components add up to 4.73 km and the y-component is 1.0 km, giving a final distance of about 4.8 kilometers. In Sample Problem 2, an airplane is spotted 215 km away at 22 degrees east of due north. By resolving the displacement vector, we find the plane is approximately 81 kilometers east and 199 kilometers north of the airport, showing how components reveal exact positions. Finally, unit vectors are special vectors with a magnitude of exactly one, used purely to indicate direction along each axis. The unit vectors i-hat, j-hat, and k-hat point along the positive x, y, and z axes respectively, forming the foundation of a three-dimensional coordinate system.