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.

A classical example of a vector spaces is $\mathbb{R}^n$, the $n$-tuples of real numbers (with the usual addition and scalar multiplication).

Elements of these spaces are often referred to as vectors.

Various vector spaces admit notions of lengths (and angles), often introduced via equipping the space with an inner product. In such a context a vector can be thought of as a quantity having a direction and magnitude.

This tag is mostly intended for questions involving vectors in a rather concrete form, e.g., finding intersection of lines and planes, determining projections, and other computations and problems involving vectors in a concrete way.

More structural and algebraic questions on vectors, are often better tagged and/or .

See here for more information.

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What is the difference between a point and a vector?

I understand that a vector has direction and magnitude whereas a point doesn't. However, in the course notes that I am using, it is stated that a point is the same as a vector. Also, can you do cross product and dot product using two points instead…
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Is there an "inverted" dot product?

The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$ What about the quantity? $$\mathbf{a} \star \mathbf{b} = \prod_{i=1}^{n}…
doc
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What does the dot product of two vectors represent?

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, $2$ and $3$, we say that it is $2$ added to itself $3$ times or something like…
Zol Tun Kul
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Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal to the plane made by $\vec a$ and $\vec b$,…
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Is arrow notation for vectors "not mathematically mature"?

Assuming that we can't bold our variables (say, we're writing math as opposed to typing it), is it "not mathematically mature" to put an arrow over a vector? I ask this because in my linear algebra class, my professor never used arrow notation, so…
hlin117
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How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\mathbf{j}$, but could not solve if I have $3$…
niko
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What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from the cosine rule. (Do correct me if I'm…
Danxe
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Why is cross product defined in the way that it is?

$\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a scalar product? So why is cross product defined…
koe
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Adding two polar vectors

Is there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?
lash
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How can I prove that 3 planes are arranged in a triangle-like shape without calculating their intersection lines?

The problem So recently in school, we should do a task somewhat like this (roughly translated): Assign a system of linear equations to each drawing Then, there were some systems of three linear equations (SLEs) where each equation was describing a…
Jonas
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Determine if vectors are linearly independent

Determine if the following set of vectors is linearly independent: $$\left[\begin{array}{r}2\\2\\0\end{array}\right],\left[\begin{array}{r}1\\-1\\1\end{array}\right],\left[\begin{array}{r}4\\2\\-2\end{array}\right]$$ I've done the following system…
Mirrana
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Difference between sum and direct sum

What is the difference between sum of two vectors and direct sum of two vector subspaces? My textbook is confusing about it. Any help would be appreciated.
Marion Crane
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What is vector division?

My question is: We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors?
Reader
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Row vector vs. Column vector

I'm a student in an elementary linear algebra course. Without bashing on my professor, I must say that s/he is very poor at answering questions, often not addressing the question itself. Throughout the course, there have been multiple questions that…
Skipher
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Cross product in higher dimensions

Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear. Suppose instead we have two vectors $x$ and $y$ in $3$-space. Then the…
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