2014年5月23日 · The dot product tells you what amount of one vector goes in the direction of another. For instance, if you pulled a box 10 meters at an inclined angle, there is a horizontal component and a vertical component to your force vector.

2017年5月8日 · Particularly, the dot product can tell us if two vectors are (anti)parallel or if they are perpendicular. We have the formula $\vec{a}\cdot\vec{b} = \lVert \vec{a}\rVert\lVert \vec{b}\rVert\cos(\theta)$ , where $\theta$ is the angle between the two …

2024年10月16日 · The dot product essentially "multiplies" 2 vectors. If the 2 vectors are perfectly aligned, then it makes sense that multiplying them would mean just multiplying their magnitudes. It's when the angle between the vectors is not 0, that things get tricky.

Re: "[the dot product] seems almost useless to me compared with the cross product of two vectors ". Please see the Wikipedia entry for Dot Product to learn more about the significance of the dot-product, and for graphic displays which help visualize what the dot product signifies (particularly the geometric interpretation). Also, you'll learn ...

$\begingroup$ @ScienceDiscoverer : I suppose because the dot product is more fundamental than trigonometry. Also because the dot product can be similarly defined in n-dimensional spaces while the trigonometry "trick" only works in two-dimensional spaces. $\endgroup$ –

Vector dot product can be seen as Power of a Circle with their Vector Difference absolute value as Circle diameter. The green segment shown is square-root of Power. Obtuse Angle Case. Here the dot product of obtuse angle separated vectors $( OA, OB ) = - OT^2 $ EDIT 3: A very rough sketch to scale ( 1 cm = 1 unit) for a particular case is enclosed.

The dot product is basically a more flexible way of working with the Euclidean norm. You know that if you have the dot product $\langle x, y \rangle$, then you can define the Euclidean norm via $$\lVert x\rVert = \sqrt{\langle x, x \rangle}.$$

The real dot product is just a special case of an inner product. In fact it's even positive definite, but general inner products need not be so. The modified dot product for complex spaces also has this positive definite property, and has the Hermitian-symmetric I mentioned above. Inner products are generalized by linear forms. I think I've ...

2024年11月22日 · According to my reference material, a dot product is the degree of "alignment" between two vectors. Consider two vectors a and b , and suppose a is projected onto b . The length of the projection of a on b , let's call it x, is a measure of the degree of alignment—because it varies with the angle between the vectors.

Geometrically the dot product of two vectors gives the angle between them (or the cosine of the angle to be precise). Algebraically, the dot product is a sum of products of the vector components between the two vectors. However, both formulae look quite different but …