All Questions
Tagged with algebra-precalculus vectors
259
questions
34
votes
3
answers
148k
views
Convert angle (radians) to a heading vector?
I have been looking everywhere trying to find out how to convert an angle in radians (expressed as -Pi to Pi) to a heading vector.
The only [x,y] answer I have ...
- 441
11
votes
1
answer
4k
views
How to define the inverse of a vector?
Most physical situations in mechanics can be modeled using a combination of derivatives - specifically, derivatives of position: velocity and acceleration. But physical situations can also be modeled ...
- 2,280
7
votes
2
answers
8k
views
Why no slope for a plane?
Similar to how lines have slopes defined in terms of $\Delta y$ and $\Delta x$, why can't planes have slopes defined in therms of $\Delta x$, $\Delta y$, and $\Delta z$?
Couldn't these could be ...
- 4,198
6
votes
3
answers
88
views
Line for set of three-dimensional vectors
If there is a set for 3D vectors $v$ where
$ v \times \begin{pmatrix} -1 \\ 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -27 \\ 8 \end{pmatrix}$
is a line, what is this line's equation?
I'm not sure ...
- 1,725
5
votes
7
answers
294
views
Why is $x_1 x_2 + x_1 x_3 + x_2 x_3$ constant for an equilateral triangle?
Consider an equilateral triangle centered at the origin of the 2D Cartesian space. Let the coordinates of its vertices be $v_1=(x_1,y_1)$, $v_2=(x_2,y_2)$ and $v_3=(x_3,y_3)$. All such triangles can ...
- 81
5
votes
2
answers
16k
views
Is there a variant of the dot-product operation that returns $\frac{a_1}{b_1} + \frac{a_2}{b_2}$ from vectors $[a_1,a_2]$ and $[b_1, b_2]$?
The dot product of $[a_1,a_2]$ and $[b_1, b_2]$ is $a_1b_1 + a_2b_2$ (and so on for bigger vectors). What I'm wondering is if there's any definition of a function s.t. the "invdot" product ...
- 1,908
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
5
votes
1
answer
772
views
Particle on vertex of a polygon moving towards adjacent particle.
Suppose we have a regular polygon with $n$ sides. On each vertex, there is a particle. Every particle moves in such a way that its velocity vector $(\vec{v})$ always points towards particle next to it....
- 7,551
4
votes
2
answers
342
views
If $\vec a,\vec b,\vec c$ be three vectors such that $|\vec a|=1,|\vec b|=2,|\vec c|=4$ and then find the value of $|2\vec a+3\vec b+4\vec c|$
If $\vec a,\vec b,\vec c$ be three vectors such that
$\vert \vec a\vert =1,\vert \vec b\vert =2,\vert \vec c\vert=4$
and
$\vec a \cdot \vec b+\vec b \cdot \vec c+\vec c \cdot\vec a=-10$
then find the ...
- 9,599
4
votes
1
answer
63
views
Why does $\vec{V_1}\times\vec{V_2}\cdot \overrightarrow{M_1M_2}\neq0$ imply that the two lines with $V_1$ and $V_2$ as direction vectors are skew?
How come that when we want to prove that two lines are skew (that is that they don't intersect nor that they are parallel) we show that $C:=\vec{V_{1}} \times \vec{V_{2}} \cdot \overrightarrow{M_{1}M_{...
- 3,231
4
votes
1
answer
224
views
Simplifying the sum $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$
How can I simplify the expression $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$?
$x$ is a vector of numbers of length $n$, and I am trying to prove that the result of the expression above is ...
- 143
4
votes
1
answer
83
views
What does the difference of constants in equations of parallel straight lines mean?
I was trying to prove the formula for distance of a point in the cartesian plane from a line. And there are many easy proofs.
I was looking for something “tastier”. For equations of planes in 3d, the ...
4
votes
1
answer
144
views
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
4
votes
1
answer
294
views
Proof that in a sequence of vectors of length N, the Nth vector must be zero
I have an assertion about a sequence of vectors which I have tested on a
computer but which I have been unable to prove. The assertion is that when
the vectors defined below are of length $N$, then ...
- 763
4
votes
2
answers
299
views
Show that any 2D vectors can be expressed in the form...
(a) Show that any 2D vector can be expressed in the form
$s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$
where $s$ and $t$ are real numbers.
(b) Let $u$ ...
- 209