| 单词 | cross product |
| 例句 | The cross product of vectors u and v is not defined. Calculus, Volume 3 2016-03-30T00:00:00Z If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. Algebra and Trigonometry 2015-02-13T00:00:00Z The book occasionally bumps up against this problem, referring the reader to earlier volumes; or Susskind might impatiently write, “If you don't know what a cross product is, please take the time to learn.” Relativity: Final ascent of physics : Nature : Nature Research 2017-09-19T04:00:00Z And most impressively, he mentioned cross product Yammer search integration, where searching from one product may reveal Yammer conversations in another. Why Microsoft's Partnership With Klout To Measure Influence At Work Is A Good Thing 2013-07-09T04:22:00Z So much for the first cross product of the dog with the jackal. The Moral and Intellectual Diversity of Races With Particular Reference to Their Respective Influence in the Civil and Political History of Mankind 2011-08-19T02:00:11.867Z If force F is acting at a distance r from the axis, then torque is equal to the cross product of r and F: τ = r × F. Calculus, Volume 3 2016-03-30T00:00:00Z We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses. Algebra and Trigonometry 2015-02-13T00:00:00Z Figure 2.58 Finding the area of a triangle by using the cross product. Calculus, Volume 3 2016-03-30T00:00:00Z The dot product of a vector with the cross product of two other vectors is called the triple scalar product because the result is a scalar. Calculus, Volume 3 2016-03-30T00:00:00Z If ‖ u ‖ = 5 and ‖ v ‖ = 2, find the angle between u and v if the magnitude of their cross product vector is equal to 9. Calculus, Volume 3 2016-03-30T00:00:00Z It is clear, for example, that the cross product is defined only for vectors in three dimensions, not for vectors in two dimensions. Calculus, Volume 3 2016-03-30T00:00:00Z In the three-dimensional case, finding the cross product of the unit tangent vector and the unit normal vector can be even more cumbersome. Calculus, Volume 3 2016-03-30T00:00:00Z From the way we have developed u × v, it should be clear that the cross product is orthogonal to both u and v. Calculus, Volume 3 2016-03-30T00:00:00Z In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculus, Volume 3 2016-03-30T00:00:00Z The cross product results in a vector, so it is sometimes called the vector product. Calculus, Volume 3 2016-03-30T00:00:00Z However, the determinant notation provides a useful mnemonic device for the cross product formula. Calculus, Volume 3 2016-03-30T00:00:00Z Let’s explore some properties of the cross product. Calculus, Volume 3 2016-03-30T00:00:00Z Unlike most operations we’ve seen, the cross product is not commutative. Calculus, Volume 3 2016-03-30T00:00:00Z For property iv., this follows directly from the definition of the cross product. Calculus, Volume 3 2016-03-30T00:00:00Z This definition of the cross product allows us to visualize or interpret the product geometrically. Calculus, Volume 3 2016-03-30T00:00:00Z Notice that this answer confirms the calculation of the cross product in Example 2.31. Calculus, Volume 3 2016-03-30T00:00:00Z We can calculate the cross product of two vectors using determinant notation. Calculus, Volume 3 2016-03-30T00:00:00Z Find unit vector w in the direction of the cross product vector u × v. Calculus, Volume 3 2016-03-30T00:00:00Z The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. Calculus, Volume 3 2016-03-30T00:00:00Z Notice that ru = 〈 0, 0, 0 〉 and rv = 〈 0, −sin v, 0 〉 , and the corresponding cross product is zero. Calculus, Volume 3 2016-03-30T00:00:00Z To use the cross product for calculating areas, we state and prove the following theorem. Calculus, Volume 3 2016-03-30T00:00:00Z Notice that each component of the cross product is positive, and therefore this vector gives the outward orientation. Calculus, Volume 3 2016-03-30T00:00:00Z We also need to find tangent vectors, compute their cross product, and use Equation 6.19. Calculus, Volume 3 2016-03-30T00:00:00Z The cross product can be used to identify a vector orthogonal to two given vectors or to a plane. Calculus, Volume 3 2016-03-30T00:00:00Z |
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