Complete the text with the words in the box:

Cross product, vector product, direction (x2), magnitude (x2), vector, specify

 

Area as a Vector

We can use the cross product to describe an area. Usually one thinks of area in terms of (1) __________ only. However, many applications in physics require that we also (2) ____________the orientation of the area. For example, if we wish to calculate the rate at which water in a stream flows through a wire loop of given area, it obviously makes a difference whether the plane of the loop is perpendicular or parallel to the flow. (If parallel, the flow through the loop is zero.) Here is how the (3) ________accomplishes this:

Consider the area of a quadrilateral formed by two (4) ________ C and D. The area A of the parallelogram is given by

A = base × height

= CD sin θ

= |C × D| .

The (5)__________of the (6) ___________ gives us the area of the parallelogram, but how can we assign a (7) ____________ to the area? In the plane of the parallelogram we can draw an infinite number of vectors pointing every which-way, so none of these vectors stands out uniquely. The only unique preferred direction is the normal to the plane, specified by a unit (8) _________ ˆn. We therefore take the vector A describing the area as parallel to ˆn. The magnitude and (9) __________of A are then given compactly by the cross product

A = C × D.

(from Kleppner D., Kolenkow R. An Introduction to Mechanics)