| 单词 | dot product |
| 例句 | Like vector addition and subtraction, the dot product has several algebraic properties. 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 As you might expect, to calculate the dot product of four-dimensional vectors, we simply add the products of the components as before, but the sum has four terms instead of three. Calculus, Volume 3 2016-03-30T00:00:00Z In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. Calculus, Volume 3 2016-03-30T00:00:00Z The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. Calculus, Volume 3 2016-03-30T00:00:00Z The fourth property shows the relationship between the magnitude of a vector and its dot product with itself: 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 The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. Calculus, Volume 3 2016-03-30T00:00:00Z Note that if u and v are two-dimensional vectors, we calculate the dot product in a similar fashion. Calculus, Volume 3 2016-03-30T00:00:00Z For this reason, the dot product is often called the scalar product. Calculus, Volume 3 2016-03-30T00:00:00Z As we have seen, the dot product is often called the scalar product because it results in a scalar. Calculus, Volume 3 2016-03-30T00:00:00Z The dot product of two vectors involves multiplying two vectors together, and the result is a scalar. Algebra and Trigonometry 2015-02-13T00:00:00Z The dot product provides a way to find the measure of this angle. Calculus, Volume 3 2016-03-30T00:00:00Z This property is a result of the fact that we can express the dot product in terms of the cosine of the angle formed by two vectors. Calculus, Volume 3 2016-03-30T00:00:00Z Direction angles are often calculated by using the dot product and the cosines of the angles, called the direction cosines. Calculus, Volume 3 2016-03-30T00:00:00Z This form of the dot product is useful for finding the measure of the angle formed by two vectors. Calculus, Volume 3 2016-03-30T00:00:00Z The dot product of two vectors is the product of the terms plus the product of the terms. Algebra and Trigonometry 2015-02-13T00:00:00Z Going back to the fruit vendor, let’s think about the dot product, q · p. Calculus, Volume 3 2016-03-30T00:00:00Z So, in this example, the dot product tells us how much money the fruit vendor had in sales on that particular day. Calculus, Volume 3 2016-03-30T00:00:00Z Remember, the dot product of orthogonal vectors is zero. Calculus, Volume 3 2016-03-30T00:00:00Z The most common application of the dot product of two vectors is in the calculation of work. Calculus, Volume 3 2016-03-30T00:00:00Z We can use the dot product to find the angle between two vectors. Algebra and Trigonometry 2015-02-13T00:00:00Z Thus, the dot product of these vectors is equal to zero, which implies they are orthogonal. Calculus, Volume 3 2016-03-30T00:00:00Z The first type of vector multiplication is called the dot product, based on the notation we use for it, and it is defined as follows: Calculus, Volume 3 2016-03-30T00:00:00Z When two vectors are combined using the dot product, the result is a scalar. Calculus, Volume 3 2016-03-30T00:00:00Z The dot product allows us to do just that. Calculus, Volume 3 2016-03-30T00:00:00Z AAA sales for the month of May can be calculated using the dot product p · q. Calculus, Volume 3 2016-03-30T00:00:00Z If we represent an applied force by a vector F and the displacement of an object by a vector s, then the work done by the force is the dot product of F and s. 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 The dot product of two vectors is the magnitude of the projection of one vector onto the other—that is, A · B = ‖ A ‖ ‖ B ‖ cos θ, where θ is the angle between the vectors. 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 Using the dot product, find the projection of vector v12 found in step 4 onto unit vector n found in step 3. Calculus, Volume 3 2016-03-30T00:00:00Z Therefore, the tangent vector and the force vector have a positive dot product all along C1, and the line integral will have positive value. Calculus, Volume 3 2016-03-30T00:00:00Z We abbreviate this “double dot product” as ∇2 . Calculus, Volume 3 2016-03-30T00:00:00Z The dot product of two vectors can be expressed, alternatively, as u · v = ‖ u ‖ ‖ v ‖ cos θ. Calculus, Volume 3 2016-03-30T00:00:00Z |
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