All Questions
Tagged with algebra-precalculus vectors
259
questions
2
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0
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46
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Is there geometric interpretation of why $\vec{v} \cdot \vec{w}=\frac{\lVert \vec{v}+\vec{w} \rVert^2 - \lVert \vec{v}-\vec{w} \rVert^2}{4}$?
Is there an interesting geometric interpretation of the relationship
$$\vec{v} \cdot \vec{w}=\frac{\lVert \vec{v}+\vec{w} \rVert^2 - \lVert \vec{v}-\vec{w} \rVert^2}{4}$$
- 5,021
0
votes
1
answer
93
views
Which law of dot product should I know in order to understand this equivalence: $ \bf v.dv =$ $d$ $(\bf v.v)$?
The context is an exercise in which it is asked to derive a well known formula regarding work :
$W=\int_{t1}^{t2} \bf F.d\bf r$ $= \frac{1}{2} mv_2^2 - \frac{1}{2} mv_1^2$
where $v_1$ and $v_2$ denote ...
- 158
1
vote
1
answer
86
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Versors (Vectors) and Trigonometry
I recently remebered, when I asked my physics high school teacher if unit vectors are somehow related to sine and cosines (or trigonometry in general). She replied to me that I was pretty lost and ...
- 775
0
votes
3
answers
397
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How to algebraically add two vectors when they are not at the origin [closed]
Suppose we have three vectors, U, V, and W. Vector U starts at (3,8) and ends at ()3,10). Vector V starts at (6,5) and ends at (7,4). Vector W starts at (9,8) and ends at (12,2). How would one go ...
-1
votes
3
answers
62
views
Why must a-b and a be on the same side of b
Question: Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $29^\circ,$ and the angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ is $84^\...
- 85
4
votes
1
answer
144
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Find $\cos\theta$ where $‖\mathbf{a}‖=6, ‖\mathbf{b}‖=8, ‖\mathbf{a}+\mathbf{b}‖=11$, and $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$. [closed]
This is a question from AOPS that I don't really understand. I would love it if someone can show me how to do this question from the very beginning.
Given vectors $\mathbf{a}$ and $\mathbf{b}$ such ...
- 85
-1
votes
2
answers
895
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Vector triple product proof
Vector triple product
How did the author arrive at the step 2 from step 1 in the above definition of a × ( b × c )?
What is your explanation?
- 1,066
0
votes
3
answers
218
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Maximum area of triangle given fixed base and perimeter
I tried solving the question in this post in another way and got a different result,
Let $0\lt a\lt b$
(i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is ...
0
votes
2
answers
75
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Show that if u · x = v · x for any vector x, then u = v.
I know that the dot product of a 2d vector (a, b) * (c, d) is equivalent to ac + bd. From this I got that u = (a, b), x = (c, d), and v = (e, f). That leads to ac + bd = ec + fd. In the end, I know I ...
5
votes
2
answers
226
views
Confused about how we scale graph axis' to make the axis' dimensionless.
I am trying to understand the solution to part $\mathrm{(iii)}$. But, for the question I'm asking to make sense I need to include the solutions to parts $\mathrm{(i)}$ and $\mathrm{(ii)}$ also:
...
- 419
0
votes
2
answers
61
views
Vectors and Planes with parametric equations
Any help would be appreciated!
Find a parametric equation of a line $L$ that is obtained as the intersection of the planes $P$ and $Q$ with scalar equations $$P: x+2y+3z=4$$ and $$Q: x-3y+z=1$$
...
- 475
0
votes
0
answers
34
views
Logic behind this rearranging of summation.
In eigenchrises video "Tensors for Beginners 1: Forward and Backward Transformations (contains error; read description!)" , at around time $6.41$ , There is a rearranging of summations I ...
- 861
1
vote
1
answer
33
views
What is the explicit expression of a plane wave in the frequency domain?
A plane wave in the time domain can be written (using notation for an electric field):
$$\boldsymbol{E}(\boldsymbol{r},t)=\boldsymbol{E}_0 e^{i(\boldsymbol{k} \boldsymbol{r}-\omega t)}$$
what is the ...
- 187
0
votes
2
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72
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Calculating $\nabla\left(\nabla \cdot \boldsymbol{r}_{0} e^{i\boldsymbol{k} \cdot \boldsymbol{r}}\right)$
In a longer derivation I ran into the following quantity:
$$
\nabla\left[\nabla\cdot\left(%
{\bf r}_{0}\,{\rm e}^{{\rm i}{\bf k} \cdot {\bf r}}\,\right)
\right]
$$
( i.e., the gradient of the ...
- 53
1
vote
0
answers
77
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Confusion with gradient and divergence of vector fields and scalar fields
In a proof, my teacher used the following assumption for an approximation:
$$
|\nabla n(\boldsymbol{r}, t)| \ll\left|\nabla \partial_{t} \boldsymbol{\tilde{r}}(\boldsymbol{r}, t)\right|
$$
where $n$ ...
- 853