What does it mean if the dot product of two vectors is negative?
Jul 27, 2018 · If the dot product gives only magnitude, then how can it be negative? For example, in this calculation: $$W = \vec{F}\cdot\vec{r} = Fr\cos\theta = (12\ \mathrm{N})(2.0\ \mathrm{m})(\cos 180^\circ) = -24\ \mathrm{N\,m} = -24\ \mathrm{J}$$
Dot Product - Formula, Examples | Dot Product of Two Vectors
The purpose of the dot product is to tell us the amount of force vector is applied in the direction of the motion vector. The dot product also lets us measure the angle that is formed by a pair of vectors and the relative position of a vector against the coordinate axes.
12.3: The Dot Product - Mathematics LibreTexts
6 days ago · The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. 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.
1.3: Dot Product - Mathematics LibreTexts
Jan 16, 2023 · By Corollary 1.8, the dot product can be thought of as a way of telling if the angle between two vectors is acute, obtuse, or a right angle, depending on whether the dot product is positive, negative, or zero, respectively.
Dot Product - Math is Fun
The Dot Product gives a scalar (ordinary number) answer, and is sometimes called the scalar product. But there is also the Cross Product which gives a vector as an answer, and is sometimes called the vector product .
10.3: The Dot Product - Mathematics LibreTexts
Dec 29, 2020 · The dot product of \(\vec u\) and \(\vec v\), denoted \(\vec u \cdot \vec v\), is \[\vec u \cdot \vec v = u_1v_1+u_2v_2.\] Let \(\vec u = \langle u_1,u_2,u_3\rangle \) and \(\vec v = \langle v_1,v_2,v_3\rangle \) in \(\mathbb{R}^3\).
calculus - Why is a negative projection vector possible?
Aug 8, 2018 · When we do this, the fact that $\cos(\theta)$ can be negative is a useful feature of the formula, not an annoyance to be worked around. We get negative values of $\mathbf a \cdot \mathbf b$ when the angle between the vectors is obtuse and only when the angle is obtuse.
What does the dot product of two vectors represent?
May 23, 2014 · If a and b point into opposite directions, i.e., 90° <θ ≤ 180°, the dot product will be the negative: a ⋅ b = − ‖p‖‖b‖. The problem is that the relationship between the dot product and the angle θ is not inherently given. By definition: a ⋅ b = ∑ i aibi.
Can dot product be negative? - California Learning Resource …
Jan 15, 2025 · Direct Answer: Yes, the Dot Product Can be Negative. The dot product of two vectors is represented by the formula: a · b = a1b1 + a2b2 + … + anbn. where a and b are two vectors in an n-dimensional space. While the sum of the products of corresponding components of the two vectors may seem straightforward, theresult can indeed be negative.
Dot Product with Projection - Math Derivations - GitHub Pages
In the above picture, the projection is positive, but if you need to "extend" $\vec v$ for drawing the line, then the projection is negative: Here's an animation showing this. The dot product is not quite the projection; it's the projection multiplied by the length of the vector being projected onto. This is the first way to think about dot ...