An Introduction to Vectors for Physics Students

Vectors for Introductory Physics Students Vectors are a mathematical concept which identifies both magnitude and direction. In Physics, vectors represent important quantities such as velocity, momentum, and force. Vectors differ from scalars (non-directional values like distance and speed) in that a scalar only gives a magnitude. For example, “an aircraft moving at 275 miles per hour” is a scalar quantity, whereas “an aircraft moving North-West at 275 miles per hour” is a vector quantity. You should be comfortable making simple calculations with vectors before taking your first Physics exam. Here are some basic facts to get you started. The variable ascribed to a vector is usually printed in bold. The magnitude of a vector V can be calculated by use of the Pythagorean theorem: V = (X2 +Y2 )1/2 where X = V cos θ Y = V sin θ The angle of the resultant vector can also be calculated by knowing inverse trigonometric functions, θ = tan-1 Y/X The sum or difference of two or more vectors is called the resultant of the vectors. The vector sum A + B is the vector joining the tail of A to the tip of B and pointing toward the tip of B. Subtracting one vector from another can be accomplished by adding a vector with equal magnitude—but opposite direction—to the first vector. This can be expressed mathematically as A – B = A + (–B), where –B represents a vector with the same magnitude as B, but pointing in the opposite direction. When a vector is multiplied by a scalar, its magnitude will change. Its direction will be either parallel or antiparallel to its original direction. If a vector A is multiplied by the scalar value n, a new vector, B, is created such that B = nA. To find the magnitude of the new vector, B, simply multiply the magnitude of A by |n|, the absolute value of n. To determine the direction of the vector B, we must look at the sign on n. If n is a positive number, then B and A are in the same direction. To generate a scalar quantity like work, we multiply the magnitudes of the two vectors of interest (force and displacement) and the cosine of the angle between the two vectors. In vector calculus, this is called the dot product (A · B): A · B = |A| |B| cos θ When generating a third vector like torque, we need to determine both its magnitude and direction. To do so, we multiply the magnitudes of the two vectors of interest (force and lever arm) and the sine of the angle between the two vectors. Once we have the magnitude, we use the right-hand rule to determine its direction. In vector calculus, this is called the cross product (A × B): A × B = |A| |B| sin θ The resultant of a cross product will always be perpendicular to the plane created by the two vectors. Another important concept is the unit vector. A unit vector is a vector of magnitude one, and can be useful in algebraic representation. Vectors will become increasingly important as you study more advanced topics in Physics. Divergence, gradient, and curl are the central focus of vector calculus and have application in the study of fluids, electrodynamics, and quantum mechanics. This is another reason why its extremely important to begin working with vectors and establishing a familiarity early in your academic career. Have suggestions about future content or need help solving a physics/math problem? 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