Questions tagged [vectors]

Use this tag for questions and problems involving vectors, e.g., in an Euclidean plane or space. More abstract questions, might better be tagged vector-spaces, linear-algebra, etc.

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A navigation problem: Is the path of the ship straight or curved?

My first post here: I’m looking for some guidance with a maths problem. A ship sets sail from England (A) to France (B) covering a distance of 20 miles at an average speed of 5mph. If the ship sails ...
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-5 votes
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How to find basis and dimensions of vector spaces

Let V={(a,b,c,d) an element of R4:b-2c+d=0}and w={(a,b,c,d) an element of R4:a=d, b=2c}. Find a basis and dimension of a. V b. W c. V intersection W
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1 vote
1 answer
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Why is the cross product not the normal in this case?

I am working through the following question: Every point x on a planar surface in three dimensions satisfies the relation $\bf{x}\cdot\bf{\hat{n}}=d$, where $\bf{\hat{n}}=a\bf{\hat{i}}+b\bf{\hat{j}}+c\...
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How Calculate To Interior Reflex Angle Of Concave

For example I have a concave polygon and I know all of coordinates of the points. How can I calculate interior reflex angle without knowing other angles ? Thanks in advance!
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Why am I getting different angles between the vectors in these two different processes?

This question came in the Dhaka University admission exam 2007-08 Question: Two vectors $\vec{P}=\hat{i}+2\hat{j}-2\hat{k}$ and $\vec{Q}=3\hat{i}+2\hat{j}+2\sqrt{3}\hat{k}$ are acting at a point at ...
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Scalars multiplied by vectors

When a positive scalar is multiplied by a vector the magnitude of the vector increases, but the direction remains constant. On the other hand when a negative scalar is multiplied by a vector the ...
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Converse of Parametric equation of plane

Suppose a plane passes through a point A ( whose position vector is a ) and parallel to two vectors b and c . Then , to any general point on plane with position vector r , I can find the equation of ...
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1 vote
2 answers
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Calculate third point of Isosceles triangle given two points and angle

In this image: (Just for annotations.. The actual triangle can be pointing to any direction) I know the coordinates of "red" base points and the "blue" vertex angle $\beta$, and I ...
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Proof of vector distributive property? [closed]

Given a vector A multiped by vector B + C, How do we prove the resultant vector is AB + AC? In addition how do we prove the distributive property for column vectors? That is, if we multiply a column ...
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-4 votes
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Sum of all possible values for $\lambda$.

If $|\vec{a}| = 1$, $|\vec{b}| = 3$, $|\vec{c}| = 4$ and $|\vec{a}-\vec{b}|² + |\vec{b}-\vec{c}|² + |\vec{c}-\vec{a}|² ≤ 12k+λ,$ and if $k$ is prime number and $λ>0$, then what is sum of all values ...
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Find the points on the curve where the tangent is vertical

Question. Given $y^{2}=x^{3}+ax+b$, find the points on the curve where the tangent line is vertical. Attempt. Let $f(x,y)=x^{3}-y^{2}+ax+b=0$ The tangent is vertical at points where the gradient is ...
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Applications of Vector Geometry

Current in maths, we are learning vector geometry. Specifically, we are trying to express lengths on a diagram ( eg. CE ) as vectors in terms of vectors that are already given. For example: But what ...
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Notation for infinite-dimensional vector output of a function $f: X \rightarrow \mathbb{R}$ on an interval

I have a function $f: X \rightarrow \mathbb{R}$. I want to create a vector of the "outputs" of the function, in the following sense: if $X$ was discrete, e.g., $X = \{x_1, x_2, \dots, x_K\}$,...
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2 votes
1 answer
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Vector Field With Elliptical Form?

I was doing some research when I found out this vector field, and I couldn't find its equations nowhere. How could I get the equation that describes this vector field, using only vector calculus and ...
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1 answer
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Convert a direction into an angle on our compass

I have a new position vector and old x and z coordinates and I need to determine the angle traveled. So far I am getting the slopes absolute value and then using that to generate an angle. Based on ...
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1 answer
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Is the dot product of vectors $u$ and $v$ the same as the dot product of $u$ and $-v$?

Can’t find this online so asking it here: Is the dot product of vectors $u$ and $v$ the same as the dot product of $u$ and $-v$? I have to use this to solve a bigger problem and stuck on this part.
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3 votes
1 answer
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Algorithm to find the angle of a direction

I need to translate the distance between two points into an angle from 0 to 359. To do this I use the new position coordinates which is defined by a vector and subtract the original position. This ...
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Linearly independence of three vectors

If I've got three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ and $\vec{a}$, $\vec{b}$ are linearly independent and $\vec{c}$ is linearly independent from $\vec{a}$, is $\vec{c}$ also linearly ...
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1 answer
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How to visualise 3D vector rotation around a line?

If we want to rotate a 2D vector we need only angle $\theta$ by which we want to rotate the vector. And we have only two possibilities: one for clockwise rotation and other for counterclockwise ...
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2 answers
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Shadow of a rod

AB is a rod which is held such that $A=(1,-2,3)$ and $B=(2,3,-4)$ . A source of light is at the origin. Find the length of the shadow of the rod on a plane screen whose equation is $x+y+2z=1$ I ...
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For which t values the two vectors are parallel to each other

$\overrightarrow{r_{1}}(t)=[t+6,-3,t+2]$ $\overrightarrow{r_{2}}(t)=[-10,t+7,-2t^{2}]$ For which t values the two vectors are parallel to each other? My try: I tried cross product, and got : $$\left[...
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How to find one vector from scalar product of two vectors

According to the dxiv's comment, to make it clear what I'm asking, I'll add a few things. I am not interested in the cross-product term in the description below. That part was clarified by the ...
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Computing a norm of a vector, induced by an inner product [closed]

Compute the norm of $\begin{bmatrix}4\\3\end{bmatrix}$, induced by the following inner product $$\langle x\mid y\rangle = 3x_1y_1 - x_1y_2 - x_2y_1 + 5x_2y_2$$ $$\begin{bmatrix}3 & -1\\-1 & 5\...
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-1 votes
0 answers
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Prove the formula for projection of vector onto vector [closed]

I am suppose to present a proof of vector onto vector formula to a group of high schoolers. Which of the proofs would you go with ?
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1 answer
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Expansion of determinants.

Could someone please explain how to use determinants in vector mathematics in detail? \begin{pmatrix}î&ĵ&\hat{k}\\1&2&3\\4&5&6\end{pmatrix} Like in this example, I always ...
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2 answers
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How to find the new speed a vector needs to be travelling at?

I'm currently working on an investigation about applying velocity and position vectors to naval ships and aircraft (In a 2-dimensional plane). In my current question, I am tracking the motion of a ...
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0 votes
1 answer
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Showing the existence of starting two vectors satisfying the below conditions for cardinality 4,5

Given two distinct nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in 3 dimensions, define a sequence of vectors by $$ \mathbf{v}_{n+2}=\mathbf{v}_{n} \times \mathbf{v}_{n+1}\left(\text { so } \...
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-2 votes
2 answers
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Finding a spanning set of a null space [closed]

$$ A= \begin{bmatrix} -3& 6 &-1& 1 &-7\\ 1 &-2& 2& 3&-1\\ 2&-4& 5& 8& -4 \end{bmatrix} $$ Please I have a problem finding the spanning set of a null ...
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0 votes
1 answer
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What is the notation of length of vector?

I am looking for a symbol to express the length of a vector or table, For example we have a vector V, len(V)==5 ( for example). How can I represent it by a symbol ?
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Find general and parametric equations of the plane containing the points $A(3, 0, 0), B(0, 1, 0)$ 'perpendicular' to the $XY-$plane.

Question : Find general and parametric equations of the plane containing the points $A(3, 0, 0), B(0, 1, 0)$ 'perpendicular' to the $XY-$plane. My Try : Seeing that the plane is perpendicular to the $...
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1 answer
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Find parameter p such that the points $ A(1,−1, 0), \ B(2 ,0 ,1), \ C(1, p, 3), \ D(2 , 2 p, 5)$ lie in the same plane. [closed]

I'm not sure how to get p with the limited amount of information i have about the system. Any suggestions on how to approach something like this?
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Order of the cross-product preference $T_u \times T_v$ vs. $T_v \times T_u$

To explain this question better, I was working through my lecture's problem sets and this problem came up: Vector Calculus 6th Edition, Anthony Tromba, Jerrold E. Marsden Consider the closed surface $...
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1 vote
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Calculating coordinates using an offset from an entity in 3D space knowing its position and rotation.

I am working on a game mod for FiveM which allows players to see bullet impacts and their angle of impact, and am trying to implement 'attaching' this evidence to objects which are not static. I have ...
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-1 votes
0 answers
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Can someone help me with this? [closed]

Find the vector form of the equation of the line in R2 that passes through P=(2,−1) and is parallel to the line with general equation 2x−3y=1.
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1 answer
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How to project a point into a new basis?

Say In $\mathbb{R}^3$ I have a point $P_{B_0} = (x, y, z)$ and its basis $B_0 = (\vec{e_0}, \vec{e_1}, \vec{e_2})$, I would like to project this point into a new basis $B_1 = (\vec{f_0}, \vec{f_1}, \...
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1 answer
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Projection of a vector on a span

I have the vectors: u=$(-2,-2,-2)$, v=$(3,-1,2)$ I have to find a vector - w=$(x,y,z)$ ( I think ) on v and u. projection of w on v is $-6v$ projection of w on u is $6u$. I got to that point: $-6|v|^2=...
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0 votes
1 answer
32 views

Finding the equation of a parabola from its graph [closed]

can chat on discord but need help asap really struguling in this class
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1 vote
0 answers
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How to show the norm of the residual in GMRES minimization problem?

I couldn't quite catch the relationship between QR and it. What I'm confused about is what S and C mean, and how can I make desired proof? Let $e_i^{n} $be the i-th unit vector in $C^n$ The solution ...
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1 answer
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How do I normalize a vector such that the sum of its squared elements is some arbitrary c?

I am trying to generate points (vectors) from the $L^2$ unit norm hypersphere uniformly at random. This post says to: Generate a random Gaussian $d$-dimensional vector $v$. Generate a random uniform ...
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-1 votes
0 answers
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calculate the mean angle between four vectors [closed]

what I am trying to to is calculate the mean angle of four normal vectors. For 2 vectors, this seems trivial. But how would i do that for 4? Thanks in advance!
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0 votes
1 answer
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Given that the two vectors 8a -b and 4a +3b are perpendicular and that |a |= 2|b |, determine the angle between a and b. [closed]

I understand you have to use the dot product rule, but I'm not sure where to take it from there.
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1 vote
1 answer
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Finding the perpendicular distance betwen two lines (3D), use of parameter

I am working on Chapter 9, Example 24, from Pearson Core Pure Mathematics book 1. 'Find the shortest distance between the parallel lines with equations 1: $r=i + 2j - k + \lambda(5i + 4j + 3k)$ and 2: ...
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Perpendicular vector in a plane in spherical coordinates system

Let's suppose we have a unit tangent vector $\mathbf{\hat{t}}$ along a curve at point $\mathbf{P}$. We can construct a plane perpendicular to this unit vector which passes through $\mathbf{P}$. I need ...
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1 vote
2 answers
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How do we simplify the formula of finding unit vector in the direction of a given vector?

Q: Find a unit vector in direction of $\vec{a}$ = 2i+2j. My solution: Now , We know unit vectors are I,j,k. So , for a vector to be a unit vector in direction of some other vector. Both vectors need ...
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Finding point on a 3-dimensional surface at a fixed given distance from a starting point [closed]

I'm in a situation in which I have a starting point of known coordinates (x, y, z) and a 3-dimensional vector starting from that point of known components (and of known length L). I would like to know ...
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0 votes
1 answer
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Optimal (in terms of remaining vector lengths) 2-dimensional projection plane of $n$ $d$-dimensional unit vectors

I have a finite number of $n$ unit vectors in $\mathbb{R}^{d}$. I would like to find a two-dimensional projection plane such that each vector has a length larger than 0 in the projection. Moreover, I ...
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1 answer
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Scalar Triple Product proof

I wanted to prove that if we change order of vectors involved in a scalar triple product in a cyclic fashion , then the product remain same . I want an elegant proof of it involving simple algebra of ...
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2 votes
1 answer
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Calculating true airspeed: Horseshoe Heading Technique

This question relates to calculating the speed of an aircraft relative to the air, based on GPS measurements (i.e. groundspeed measurements). Specifically it is about David Rogers' Horseshoe Heading ...
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2 answers
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Vector being perpendicular to a plane

Let us consider a vector $\vec P=x\hat i+y\hat j+z\hat k$. How do we test whether it is perpendicular or not to a certain plane?(We take $xy$ plane for simplicity). My take on this was this. Let us ...
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1 answer
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Did I find the vector projection correctly?

I was a bit worried if I followed all the conventions correctly or not. Let the vector projection of a vector $\vec{a}$ on a non-zero vector $\vec{b}$ is $\vec{a_1}$. My attempt: $$\vec{a_1}=\frac{(\...
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