# 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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### Given the point $T$ and the line $p$, find the distance between their orthogonal projections onto the plane $\pi$.

Let $T'$ and $p'$ be the orthogonal projections of the point $T(-8,2,-3)$ and the line $$p\ldots\frac{x}4=\frac{y-4}3=\frac{z+1}{-2}$$ onto the plane $\pi\ldots x-y+3z+8=0$. Find the distance ...
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### Eigenvectors for eigenvalue with multiplicity $\mu = 2$

I'm looking for a way to determine linearly independent eigenvectors if an eigenvalue has a multiplicity of e.g. $2$. I've looked for this online but cannot really seem to find a satisfying answer to ...
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### Find parametric equations for the midpoint $P$ of the ladder

The following problem appears at MIT OCW Course 18.02 multivariable calculus. The top extremity of a ladder of length $L$ rests against a vertical wall, while the bottom is being pulled away. ...
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### What exactly does it mean for a vector to have a direction?

Vectors are defined as having magnitude and direction. If I understand it correctly, their magnitude is their length, meaning they have the properties of a line segment. What does it mean for a vector ...
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### Can a discontinuous vector field be conservative?

I consider a closed curve within a 2-dimensional vector field, passing through two points $A$ and $B$. Going clockwise, along the path from $A$ to $B$ the field is constant $\vec{v}(\vec{x}) = \vec{k}$...
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### Points A, B and C have position vectors a = i + 2 j, b = 2i + j and c = 2i − 3 j. Find the shortest distance from B to line AC

Points A, B and C have position vectors a = i + 2 j, b = 2i + j and c = 2i − 3 j. Find: The shortest distance from B to line AC The area of triangle ABC What I have done so far: Found AB and ...
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### How to find the limit of a dot product of vectors

For a vector $v=\left( {1 \over 2},{1\over 2},{1\over 2}\right)\;$ the following is defined \begin{aligned}w_1&=(e,e+2,e-2)\\ w_n&=v\times w_{n-1}+(2,-4,2)\quad (n\geq2),\end{aligned} $e$ ...
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### A geodesic on a unit sphere

Points $A(\cos\alpha,0,\sin\alpha)$ and $B(0,\cos\beta,\sin\beta)$, $(0<\alpha$ and $\beta<\pi/2)$ are on a unit sphere and $l$ is the shortest line (geodesic) between $A$ and $B$ on the sphere. ...
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### unit vector conceptual

For the following, is it always going to be equal to 1? $$\left|\frac{\vec{r}}{\vec{r}}\right|$$
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### Verify that a mapping onto a subspace is a projection mapping [closed]

$V$ is a vector space with basis $\{a_1,a_2, \dots a_n\}$ and $W$ is its vector subspace with basis $\{b_1,b_2, \dots,b_k\}$. EDIT - $\{b_1,b_2, \dots,b_k\}$ is a subset of $\{a_1,a_2, \dots a_n\}$....
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### Find the ratio in which edges are divided in a tetrahedron

In a regular tetrahedron, E and F are the midpoints of edges AD and BC, respectively.M and N are on segments CD and EF, respectively, such that α=∠MNC=π/4, β=∠NME=π/3. In which ratio are EF and CD ...
I have the vector $\vec c$ that is: $\vec c= \frac{\sum_{i=1}^n m_i\vec r_i}{\sum_{i=1}^n m_i}$ where $\vec r_i$ is a vector and $m_i$ is a scalar I need to proof the folowwing equality for any ...