VectorComponents

　　The basis vectors are important because you can express any vector in terms of the sum of multiples of the two basis vectors. For instance, the vector (3,4) that we discussed above—call it A—can be expressed as the vector sum .

　　The vector　is called the “x-component” of A and the　is called the “y-component” of A. In this book, we will use subscripts to denote vector components. For example, the x-component of A is　and the y-component of vector A is .

　　The direction of a vector can be expressed in terms of the angle　by which it is rotated counterclockwise from the x-axis.　　Vector Decomposition　　The process of finding a vector’s components is known as “resolving,” “decomposing,” or “breaking down” a vector. Let’s take the example, illustrated above, of a vector, A, with a magnitude of A and a direction　above the x-axis. Because , , and A form a right triangle, we can use trigonometry to solve this problem. Applying the trigonometric definitions of cosine and sine,

　　we find:

　　Vector Addition Using Components　　Vector decomposition is particularly useful when you’re called upon to add two vectors that are neither parallel nor perpendicular. In such a case, you will want to resolve one vector into components that run parallel and perpendicular to the other vector.　　Example

　　Two ropes are tied to a box on a frictionless surface. One rope pulls due east with a force of 2.0N. The second rope pulls with a force of 4.0N at an angle 30 o west of north, as shown in the diagram. What is the total force acting on the box?

　　To solve this problem, we need to resolve the force on the second rope into its northward and westward components.

　　Because the force is directed 30o west of north, its northward component is　　and its westward component is

　　Since the eastward component is also 2.0N, the eastward and westward components cancel one another out. The resultant force is directed due north, with a force of approximately 3.4N.　　You can justify this answer by using the parallelogram method. If you fill out the half-completed parallelogram formed by the two vectors in the diagram above, you will find that the opposite corner of the parallelogram is directly above the corner made by the tails of those two vectors.

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