All Questions
Tagged with algebra-precalculus vectors
259
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I don't understand how difference of vectors work {HOMEWORK} [duplicate]
So in the picture we have vectors u and v. Our goal is to find $vβu$
From what I know, the subtraction of vectors is just reversing the direction of the $2^{nd}$ vector & then finding the ...
- 59
1
vote
1
answer
270
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Sum of the vectors from centre $O$ to the polygon vertices
I'm attempting to calculate the sum of the vectors from the center of a regular polygon to each of the vertices. I have already solve it in a complex analysis manner:
To represent the vertices of a ...
- 23
0
votes
0
answers
24
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How do I prove that the angle between two 2d vectors depends of sign of dot product of two 2D.?
How would you prove that given two 2D vectors in the $\vec{v} = \begin{bmatrix}
v_{1} \\
v_{2} \\
\end{bmatrix}$ and $\vec{u} = \begin{bmatrix}
u_{1} \\
...
- 11
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votes
0
answers
28
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Determination of a positive basis
In 1-4 of Do Carmo's Curves and Surfaces, he states that so long as $\mathbf{u} \land \mathbf{v} \neq 0$ for two vectors $\mathbf{u}$ and $\mathbf{v}$ (where $\land$ denotes the cross product between ...
- 111
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1
answer
48
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Proof that two matrices are row-equivalent iff they have the same nullspace
The matrices are both of size m x n over some field F, obviously.
The first direction of this proposition is clear enough, however the opposite direction (same nullspace -> row-equivalence) is ...
- 351
-2
votes
2
answers
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Advanced Algebra Problem Maybe linked with Vectors? [duplicate]
$x^2 +y^2 + xy = 25$
$y^2 + z^2 + yz = 49$
$z^2 + x^2 + zx = 64$
Find $(x + y + z)^2 -100$
Here's My Approach :
$x^2 + y^2 -2xycos120 = 25$. This Equation looked too similar to the subtraction of the ...
- 59
5
votes
7
answers
294
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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 ...
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votes
1
answer
131
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Do the BEDMAS rules apply to different types of mathematical objects, such as matrices or vectors?
I know that the BEDMAS rules (Brackets - Exponents - Division OR Multiplications - Addition OR Subtraction) for Order of Operations apply to scalars and algebraic expressions.
Do the BEDMAS rules for ...
- 21
3
votes
1
answer
133
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Equation of plane $\mathbf{x} = (1, 0, 1) + s(1, 3, -1) + t(2, 2, 1)$
I'm given that the plane $W$ in $\mathbb R^3$ can be written as
$$W: \mathbf{x} = (1, 0, 1) + s(1, 3, -1) + t(2, 2, 1)$$
where $s$ and $t$ are real numbers.
My task is to write $W$ as a general ...
- 39
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0
answers
210
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Let $|\vec a|=|\vec b|=2$ and $|\vec c|=1$.Find the difference between maximum and minimum possible values of $|\vec a+\vec b|$
Let $|\vec a|=|\vec b|=2$ and $|\vec c|=1$. Also $(\vec a-\vec c)\cdot(\vec b-\vec c)=0$.
Find the difference between maximum and minimum possible values of $|\vec a+\vec b|$
My Attempt
$|\vec a+\vec ...
- 9,599
-2
votes
1
answer
50
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Why can the dot product of two vectors be written in the form $a_x b_x + a_y b_y + a_z b_z$ [closed]
My intuition says that $(a_x + a_y +a_z) \cdot (b_x + b_y + b_z) $ would expand out to be a quadratic... Why isn't this the case?
1
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1
answer
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How is this equation valid$?$
If $$\vec aΓ\vec r=\vec b+t\vec a$$ and $$\vec a \cdot\vec r =3$$ where $\vec a =2\hat i+\hat j-\hat k$ and $\vec b=-\hat i-2\hat j+\hat k$ then find $\vec r$.
I have found the $\vec r$.
My question ...
2
votes
1
answer
95
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Using vectors, find the rate of change of distance between two particles.
Particle A moves along the positive x-axis, and particle B along the line $$y=-\sqrt{3}x$$ for $x\in\left(-\infty,0\right]$ where $x$ and $y$ are in meters. At a certain time, $A$ is at the point $\...
- 164
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1
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49
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Determining the centers of two circles at the moment of contact
This problem appeared in a coding project im in the middle of and its driving me crazy. The problem is as follows:
Assume there are two circles $c_1$ and $c_2$, with known position vectors $p_1$, $p_2$...
0
votes
1
answer
34
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Average angular velocity of a particle that retrogrades based on position of another
I originally thought that this problem would be trivial to solve, but it has proven more difficult than I expected.
Suppose there are two runners on a circular track with radius r. Runner a runs at a ...
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 βtastierβ. For equations of planes in 3d, the ...
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1
answer
77
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Finding directional angle of vector in $\mathbb{R}^2$
I'm not sure if this is correct or not and need someone to check.
I have a vector $\vec{v} = 4\left(\frac{-1}{2}, 1\right) - \frac{1}{2}(4, 8)$
I simplified it to $(-4, 0)$
So the directional angle ...
- 23
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1
answer
71
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Projectile Vectors Speed
At time $t = 0$, a projectile of mass $m$ is launched from the origin at an angle $Ξ±$ to the
horizontal with speed $U$.
Let the position vector of the projectile be $\mathbf{r} = x\mathbf{i}+z\mathbf{...
user1071088
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Recreational Math Problem about unknown amount of rotating vectors
I created this recreational math problem that where it looks like solver doesn't have enough information to solve it. This kind of algebra problem I haven't really seen in the wild before, as you don'...
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2
answers
342
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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
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vote
0
answers
54
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An exercise about vector multiplication/scalar product
I came across an exercise in vector multiplication but couldnβt find the formal answer for it to check myself.
I would really appreciate if someone could show me the correct answer for this exercise.
...
- 21
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votes
2
answers
54
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Finding the equation of a parabola from its graph [closed]
can chat on discord but need help asap really struguling in this class
- 11
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0
answers
195
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Find a formula for the magnitude of an arbitrary linear combination ax+by |||| Find the precise condition on $a, b, c$ and $d$ under which the lin....
Let x and y be two perpendicular unit vectors.
(a) Find a formula for the magnitude of an arbitrary linear combination ax+by of x and y in terms of a and b.
(b) Find the precise condition on a,b,c and ...
- 11
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votes
1
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41
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Prove that the addition of two unit vectors bisect the angle between the vectors themselves
Here is a picture for clarity:
So here is what I attempted:
$a \dot{} b = |a||b|\cos(\theta)$
$a \dot{} (\hat{a}+\hat{b})$ $= |a||\hat{a}+\hat{b}|\cos(\gamma)$
$a \dot{}$ $(\frac{a}{|a|} + \frac{b}...
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Problem with solving simultaneous equations
In part v of part a of the question, they asked me to find the coordinates of point P. I know that:
Also note that part v of question relates back to part iv(for context)
$|OP| = |AP| = |BP| = 5\sqrt{...
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0
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273
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Find the precise condition on a,b,c, and d under which the linear combinations ax+by and cx+dy are perpendicular.
"Let $\mathbf{x}$ and $\mathbf{y}$ be two perpendicular unit vectors.
(a) Find a formula for the magnitude of an arbitrary linear combination
$$a\mathbf{x} + b\mathbf{y}$$ of $\mathbf{x}$ and $\...
- 31
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vote
1
answer
259
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What does being proportional mean?
Question:
Two concurrent forces act along the sides CA and CB of a triangle. Their magnitudes are proportional to $\cos (A)$ and $\cos (B)$ respectively. Prove that their resultant is proportional to $...
- 3,186
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votes
3
answers
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I don't fully understand why Pythagorean theorem works with velocity vectors.
I get why it works with displacement because that's what the theorem was originally meant for, lengths.... I find it harder to wrap my head around it when its velocity. If anyone has a good ...
-1
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2
answers
78
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Why is a+b+c = a-a+c? [closed]
Why does $a+b+c = a-a+c$? I don't understand. Is it some math property that i didn't know of?
- 3
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1
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How do we know the position of fixed point in this Q?
Q: A particle moves on a given straight line with a constant speed v. At a certain time it is at a point $P$ on its straight line path. $O$ is a fixed point. Show that (OPΓv)is independent of the ...
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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
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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 ...
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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
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3
answers
62
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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^\...
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1
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Find $\cos\theta$ where $β\mathbf{a}β=6, β\mathbf{b}β=8, β\mathbf{a}+\mathbf{b}β=11$, 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 ...
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votes
2
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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 Γ ( b Γ c )?
What is your explanation?
- 1,066
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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
views
Show that if u Β· x = v Β· 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 ...
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2
answers
226
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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:
...
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votes
2
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61
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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 $$Q: x-3y+z=1$$
...
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0
answers
34
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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 ...
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vote
1
answer
33
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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 ...
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votes
2
answers
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
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votes
1
answer
273
views
Find vectors $u$ and $v$ such that the parametrization
Find vectors $u$ and $v$ such that the parametrization $w(t) = u + tv$ describes the line containing the points $(5,2)$ and $(-1,3)$.
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1
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1
answer
300
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Finding $k$ such that $\binom{-2}{k}$ is the direction vector of the line $y=\frac17(4x+1)$
I'm new here, I asked my friend about this question and he told me to go on this math forum. If someone can get me started on these questions it would make my day. Thank You
Find the value of $k$ ...
- 19
0
votes
2
answers
492
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Really confused with unit vectors
Let $\mathbf{u}$ and $\mathbf{v}$ be linearly independent unit vectors. Find the set of all possible values of $\mathbf{u}\cdot\mathbf{v}$. Give your answer in interval notation.
What is the maximum ...
user834359
1
vote
3
answers
71
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Line along shortest distance between two skew lines
We have points $A(1,-1,1),B(6,-3,1),C(2,-1,5),D(5,1,1)$
We also have a line $p$ that goes through the edges $AD$ and $BC$ perpendicularly.
Find the equation of this line:
My try:
$$AD = (4,2,0)$$
$$BC ...
- 2,527
-2
votes
4
answers
37
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Find $π₯,π¦,π§$ such that $(π₯βπ¦, π₯+π¦, π§β1)=(4, 2, 3)$. [closed]
I am stuck at this question can someone help me?
I've been trying it all morning but I just cant quite get it going
maybe one of the lads can help
