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.

12538 questions
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how does the dot product determine similarity?

I want to know how the dot product can determine whether two vectors are similar? I know that the formula $$\cos(\theta) = \frac{u \cdot v }{ ||u||\,||v||}$$ means something, but don't know what.
FJam
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Proof that the cross product is not associative without using components

I need to show that the cross product is not associative without using components. I understand how to do it with components, which leads to an immediate counterexample, but other than that I am not sure how to do it.
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Squaring a Vector?

So this one is basic. And should be pretty quick. Lets say that I have a vector $\vec{r}$: $\vec{r} = \vec{x} + \vec{y} + \vec{z}$ Is this true: $\vec{r}^{2} = \vec{x}^{2} + \vec{y}^{2} + \vec{z}^{2}$ I know that you can't really multiply a vector…
16
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3 answers

How to find the vector formula for the bisector of given two vectors?

There are two vectors called $\vec{a}$ and $\vec{b}$. Vector $\vec{c}$ is the bisector and it can be given as $$\vec{c} = |b|\vec{a} + |a|\vec{b}$$ How to prove that? I have used the dot product method. But there I can't find the angle between them.…
user228285
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14
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What's the best way to find a perpendicular vector?

Let's say I have a vector like this: $⟨-2,7,4⟩$ What's the best way to find a perpendicular vector for this? Right now I'm doing $-2x+7y+4z=0$ And plugging in random values for $x$, $y$, and $z$ until I get $0$. This can't be right, right?
Si Random
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12
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2 answers

How to resize a vector to a specific length?

I have a 3D vector u defined through two points A (0/2/0) and B (3/3/3). u = [3/1/3] u.length = sqrt(3²+1²+3²) = 4.24... How can I get a new vector v with v.length = 1.5 which has the same origin A and direction as u?
VoidCatz
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11
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A vector is defined to have a magnitude and *a* direction, but the zero vector has no *single* direction. So, how is the zero vector a vector?

The popular definition of a vector is A vector is an object that has both a magnitude and a direction. We know that zero vector has no specific single direction. Then how can it be a vector?
hanugm
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11
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Difference between collinear vectors and parallel vectors?

I can't understand the difference between the two. The definitions are as written in textbook: Parallel vectors are vectors which have same or parallel support. They can have equal or unequal magnitudes and their directions may be same or…
11
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5 answers

Proof for parallelogram law of vector addition

The Statement of Parallelogram law of vector addition is, If two vectors are considered to be the adjacent sides of a parallelogram, then the resultant of two vectors is given by the vector that is a diagonal passing through the point of contact of…
10
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8 answers

why do we use cosine as the expression of vector dot product?

When we do vector products, we use two different methods. One is the vector dot product, another is vector cross product. The equation of the vector dot product is $$\textbf A \cdot \textbf B =|\textbf A| | \textbf B| \cos\theta,$$ where $\theta$ is…
Shams Tarek
9
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6 answers

Origin of vectors

Background I was reviewing notes of physics, and i realized that something about the mathematics of vectors was wrong in my head. Example-problem Suppose a vector is $A=5\textbf{i} + 3\textbf{j}$, and other $B=7\textbf{i}+3\textbf{j}$. Then…
user436603
8
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2 answers

How to prove invariance of dot-product to rotation of coordinate system

Using the definition of a dot-product as the sum of the products of the various components, how do you prove that the dot product will remain the same when the coordinate system rotates? Preferably an intuitive proof please, explainable to a…
Kevin
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8
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2 answers

100 dot product riddle

I was given this riddle by one of my professors, and was wondering if anyone could give some hints on this problem. Say I have 100 vectors, $x_1, x_2, ... x_{100}$. I compute every dot product pair, excluding self-pairs, so a vector is not dotted…
8
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3 answers

Why is my alternate method of calculating scalar products not working?

The exercise is such: Given that $|\vec{a}| = 3$, $|\vec{b}| = 2$ and $\varphi = 60^{\circ}$ (the angle between vectors $\vec{a}$ and $\vec{b}$), calcluate scalar product $(\vec{a}+2\vec{b}) \cdot (2\vec{a} - \vec{b})$. My initial thought was to…
God bless
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8
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Hairy ball theorem: Do the poles have to be opposite each other?

Many graphical examples of the hairy ball theorem show the cowlicks at opposite poles. Is this arrangement necessary, or arbitrary? I believe technically the theorem says how many cowlicks there must be, not where they are. Intuitively, it seems…
Superbest
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