May 23, 2014 · 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.

May 8, 2017 · 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 …

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 ...

Oct 16, 2024 · 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.

$\begingroup$ @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.

Nov 22, 2024 · 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.

Oct 2, 2015 · Depending how you chose the axis the vector $\vec v$ can be in any quadrant and the kay fact that determine the sign of the dot product is the angle between $\vec v$ and $\vec w$. $\endgroup$ – Emilio Novati

Dec 12, 2014 · As described above in comments, dot product and scalar projection are same when projecting onto a unit vector. When we're projecting a vector b on a non-unit vector, here a, then scalar projection finds respective unit vector (normalize a by dividing by norm ||a||) and projects on the unit vector. However, dot product doesn't do this normalization.

Jun 1, 2020 · Dot product between sum and difference between two vectors. 1. 3 Vectors dot product orthogonal. 6 ...