Note that is not a usual polar vector , but has slightly different transformation properties and is therefore a so-called pseudovector Arfken , pp. Jeffreys and Jeffreys use the notation to denote the cross product. The cross product is implemented in the Wolfram Language as Cross [ a , b ]. A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?
Another joke presented on the television sitcom Head of the Class asks, "What do you get when you cross an elephant and a grape? In two dimensions, the analog of the cross product for and is. Arfken, G. Orlando, FL: Academic Press, pp. Jeffreys, H. Cambridge, England: Cambridge University Press, pp. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. The corkscrew right-hand rule is a common mnemonic used to determine the direction of the vector product.
The direction of the cross product is given by the progression of the corkscrew. The mechanical advantage that a familiar tool called a wrench provides Figure depends on magnitude F of the applied force, on its direction with respect to the wrench handle, and on how far from the nut this force is applied.
To loosen a rusty nut, a Find the magnitude and direction of the torque applied to the nut. The magnitude of this torque is. Physically, it means the wrench is most effective—giving us the best mechanical advantage—when we apply the force perpendicular to the wrench handle. In this way, we obtain the solution without reference to the corkscrew rule.
Similar to the dot product Figure , the cross product has the following distributive property:. The distributive property is applied frequently when vectors are expressed in their component forms, in terms of unit vectors of Cartesian axes. We can repeat similar reasoning for the remaining pairs of unit vectors. The results of these multiplications are. The cross product of two different unit vectors is always a third unit vector.
When two unit vectors in the cross product appear in the cyclic order, the result of such a multiplication is the remaining unit vector, as illustrated in Figure b. When unit vectors in the cross product appear in a different order, the result is a unit vector that is antiparallel to the remaining unit vector i. In practice, when the task is to find cross products of vectors that are given in vector component form, this rule for the cross-multiplication of unit vectors is very useful.
These products have the positive sign. These products have the negative sign. We can use the distributive property Figure , the anticommutative property Figure , and the results in Figure and Figure for unit vectors to perform the following algebra:.
When performing algebraic operations involving the cross product, be very careful about keeping the correct order of multiplication because the cross product is anticommutative. The last two steps that we still have to do to complete our task are, first, grouping the terms that contain a common unit vector and, second, factoring.
In this way we obtain the following very useful expression for the computation of the cross product:. In this expression, the scalar components of the cross-product vector are. When finding the cross product, in practice, we can use either Figure or Figure , depending on which one of them seems to be less complex computationally. They both lead to the same final result. One way to make sure if the final result is correct is to use them both.
When moving in a magnetic field, some particles may experience a magnetic force. To compute the vector product we can either use Figure or compute the product directly, whichever way is simpler. Hence, the magnetic force vector is perpendicular to the magnetic field vector. We could have saved some time if we had computed the scalar product earlier. Even without actually computing the scalar product, we can predict that the magnetic force vector must always be perpendicular to the magnetic field vector because of the way this vector is constructed.
The dot product is a scalar; the cross product is a vector. Later chapters use the terms dot product and scalar product interchangeably. Similarly, the terms cross product and vector product are used interchangeably. How can you correct them? If the cross product of two vectors vanishes, what can you say about their directions?
If the dot product of two vectors vanishes, what can you say about their directions? What is the dot product of a vector with the cross product that this vector has with another vector? Why or why not? You fly [latex] Find y and r. Starting at the island of Moi in an unknown archipelago, a fishing boat makes a round trip with two stops at the islands of Noi and Poi. It sails from Moi for 4. Looking at the above graph, you can use the right-hand rule to determine the following results. In particular, the cross product of any standard unit vector with itself is the zero vector.
It obeys the following properties:. These properties mean that the cross product is linear. We can use these properties, along with the cross product of the standard unit vectors, to write the formula for the cross product in terms of components. Then, the manipulations are much easier.
However, it's just a matter of repeating the same manipulations above using the cross product of unit vectors and the properties of the cross product.
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