Use this tag for questions involving vectors, quantities that have magnitude and direction.

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How can Ishow that $(\vec a \times \vec b) \cdot (\vec a \times \vec b)=|\vec a|^2|\vec b|^2-(\vec a \cdot \vec b)^2$ using index notation?

I'm trying to use index notation (i.e. Einstein summation notation) in order to show that $(\vec a \times \vec b) \cdot (\vec a \times \vec b)=|\vec a|^2|\vec b|^2-(\vec a \cdot \vec b)^2$. Here's ...
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0answers
4 views

Show that an affine combination can be written as a point plus a vector?

Show that an affine combination can be written as a point plus a vector:
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2answers
20 views

Parametric / vector question.

Question 10 [10 points] Let L be the line with parametric equations $$ x = −6−3t $$ $$ y = 6+3t $$ $$ z = −8+2t $$ Find the vector equation for a line that passes through the point P=(−1, 2, 3) and ...
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1answer
26 views

Prove True or false : If A and B are nxn invertible matrices and (AB)^2=A^2B^2, then AB=BA

This looks like it is false but the thing is I can't find a counter example for it.
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0answers
25 views

linear algebra find a line that intersects another line

question: Let L be the line with parametric equations x = 3+2t y = −5 z = −6−3t Find the vector equation for a line that passes through the point P=(−5, 5, −6) and intersects L at a point that is ...
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2answers
31 views

Why does the cross product follow the right hand rule?

I know the cross product follows the RHR. I know it has to be normal to the plane defined by both vectors, and can prove this. But I can't prove it obeys the RHR. I saw in another post that it is ...
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0answers
10 views

verifying the geometric relationship in the Blinn-Phong reflection model.

I would like to prove to myself that the angle between the normal and the half vector is twice the angle between the reflection vector and the eye vector, as shown in this image taken from the ...
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1answer
9 views

Finding the set of values for the parameter of a line such that the perpendicular distance from the line to a plane is less than 4

The plane $Π_1$ has equation $$r=i+2j+k+ \theta (2j-k) + \phi (3i+2j-2k) $$ Which is $$2x+3y+6z=14$$ The line $l$ has equation $$r = 3i + 8j + 2k + t(4i + 6j + 5k)$$ The ...
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2answers
17 views

Normal vectors and tangent planes

Could you check my work please? Let me know if it's right or wrong. We have the level surface $$f(x, y, z) = xyz -6$$ The normal vector is equal to the gradient, so at the point $(a, b, c)$ $$\nabla ...
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0answers
13 views

Finding equation of plane in 3D

I was given 3 points on a plane: (5, 4, −8),(1, 6, −3) and (7, −2, 5) I was trying to find the equation of the plane and did the following: I chose two vectors to cross multiply to find the normal ...
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0answers
19 views

plane generated by n linearly independent n-dimensional vectors

Prove that the following statement is true. I'm not sure whether the term 'linear combination (narrow sense)' is widely used since I'm studying in Korea. According to my professor, the term ...
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0answers
9 views

Clarifications regarding matrix transformations.

I have an equation which looks like this: Pos1 * L1 * X * L2 = Pos2 * R1 Where Pos1 and Pos2 are vectors. L1,X,L2 and R1 are matrices. I have to find the value for the matrix X. Please let me know ...
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3answers
19 views

Equation for a plane perpendicular to a line through two given points

The following type of question is quite popular with examiners at the institution where I study. Find an equation of the plane containing the point $(0, 1, 1)$ and perpendicular to the line passing ...
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0answers
20 views

How would I find the normal vector to $xyz=1$ [on hold]

How would I find the normal vector at the point $(a, b, c)$? Is it $$ \left( \frac{1}{p}, \frac{1}{q}, \frac{1}{r}\right) $$
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0answers
21 views

How would I find the normal vector to the level surface $xyz=1$ [on hold]

At the point $(p, q, r)$ would the normal vector be $$ \left( \frac{1}{p}, \frac{1}{q}, \frac{1}{r}\right) $$
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2answers
43 views

Form a basis for R^3? [on hold]

This is a homework problem and I need help on. Consider the matrix with the given vectors as its columns. Do (1, -1, 3), (-1, 5, 1), (1, -3, 1) form a basis for R^3?
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2answers
18 views

Converting plane equation from $ax+by+cz=d$ to $r=a+\lambda b+\mu c$

The equation of the plane Π is $$2x + 3y + 4z= 48$$ Obtain a vector equation of Π in the form $r = a + λb + μc$, where a, b and c are of the form pi, qi + rj and si + tk respectively, and ...
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1answer
25 views

Prove True or false

if the rref of a has a row of 0', then the set of row vectors of a is linearly dependent. Please help me prove or give a counterexample
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2answers
18 views

Find 2 unknown scalars that makes two vectors equal to this vector. [on hold]

Let u = (1, -1 , 3, 5) and v = (2, 1, 0, -3) Find scalars a and b so au + bv = (1, -4, 9, 18) Please keep it simple! thanks.
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1answer
29 views

Linearly Independent or Dependence of Complex Vectors — Homework Help [closed]

Determine whether the indicated sets of complex vectors are linearly independent or dependent. $\left[\begin{array}{cc}i\\1\end{array}\right]$$\left[\begin{array}{cc}1\\i\end{array}\right]$ ...
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0answers
41 views

How to show that e.g. $E(\mathbf{w}) = \ldots \Rightarrow \frac{1}{2}(\Phi\mathbf{w} - \mathbf{t})^T(\Phi\mathbf{w} - \mathbf{t}) $

We have to show for a few formulas that they can also be written in matrix notation. For example: For $\mathbf{x}=(x_0,x_1,\ldots,x_n)$,xi∈R $\sum_{n=1}^ix_n^2 = ...
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0answers
20 views

Excercise: Find the volume of the parallelepiped

Find the volume $V$ of the parallelepiped whose four adjacent vertices are the points: $A = (−2, 1, 0)$, $B = (2, 3, 2)$, $C = (1, 4, −1)$, and $D = (3, 6, 1)$. I know how to find it with three ...
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1answer
22 views

Instead of mid-point, how do i find a third of the way up the line instead.

I'm working with the program Maya, and i need to script a way to find various fractions up a line for each 3D vector. For example, lets say I have the vector (0, 0, 0), and the vector (12, 12, 12). ...
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2answers
19 views

Vector proof that $d_1^2 + d_2^2 = 2a^2 + 2b^2$ in a parallelogram

How would one prove the equality of the sum of squares of diagonals and twice the sum of squares of the two sides: $$\left|\mathbf{p} + \mathbf{q}\right|^2 + \left|\mathbf{p} - \mathbf{q}\right|^2 = ...
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1answer
20 views

How to sort vertices of a polygon in counter clockwise order?

How to sort vertices of a polygon in counter clockwise order? I want to create a function (algorithm) which compares two vectors $\vec v$ and $\vec u$ which are vertices in a polygon. It should ...
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1answer
24 views

How can I show that $(a^\top \otimes bb^\top \otimes a) = (a \otimes b)(b \otimes a)^\top$?

Let symbol $\otimes$ denotes the Kronecker product, $a \in \mathbb R^n$ and $b \in \mathbb R^m$. How can I show that $(a^\top \otimes bb^\top \otimes a) = (ba^\top \otimes ab^\top)$ ? My final goal ...
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0answers
11 views

Perpendicular rays

I have a ray $r$ and a ray $v$ (I know both the start point and the direction). I am trying to find a ray $g$ with the following constraints: $g$ is perpendicular to $v$, i.e. $g \cdot v = 0$ $g$ is ...
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0answers
29 views

A counter example of the direct sum of sub spaces.

I was asked to give examples of 3 subspaces where W + V + U is not the direct sum of these 3 subspaces. W, V, and U are subspaces of a vector space, just to clarify. I am having trouble finding out ...
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30 views

The meaning of cross product

i think i understand the cross product ( how much of vector A in the direction of vector B = how much two vectors are parallel to each other ) correct me if I'm wrong . now when i come to cross ...
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2answers
24 views

Perpendicular to a plane given a parametric equation

i'm having a small issue with a certain question. Given a parametric equation of a plane $x=5-2a-3b$, $y=3-4a+2b$, $z=7-6a-2b$, find a point $P$ on the plane so that the position vector of $P$ is ...
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1answer
17 views

How to find line parallel to direction vector and passing through a specific point?

I am give the point $(1,0,-3)$ and the vector $2i-4j+5k$ Find the equation of the line parallel to vector and passing through point $(1,0,-3)$ Could one use the fact that the dot product between the ...
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1answer
10 views

Converting to Spherical Coordinates that have a Large Azimuth?

I've run into a problem converting Cartesian coordinates to spherical coordinates. Say I've got a vector/point $p=(-1,5,7\frac{2}{3})$. Obviously, finding the polar angle/inclination isn't going to ...
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2answers
29 views

How: $ (\vec b + \vec c)(\vec b - \vec c) = 0 \implies \frac{\vec b + \vec c}{2}.(\vec c - \vec b) = 0 $

I am having difficulty understanding one step of the solution to a question. question:- Using vectors, prove that the median to the base of an isosceles triangle is perpendicular to the base. ...
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3answers
59 views

Is $\nabla\cdot{F} = F\cdot\nabla$?

According to the vector dot product, $a\cdot{}b = b\cdot{a}$ for all $a, b.$ However, is $\nabla\cdot{F} = F\cdot\nabla$ (where $\nabla\cdot{F} = \operatorname{div} F$)?
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1answer
35 views

How to calculate the angle between 2 vectors in 3D space given a preset function

In my application, I am attempting to connect 2 points in 3d space with a cylinder via a function taking in 2 vectors. I understand that I need the angle to apply to the cylinder. As I understand, I ...
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1answer
22 views

Finding the curl of a cross product

Let $\mathbf{x}$ be the position vector, $\mathbf{a}$ be a constant vector. I need to show that: $$\text{curl}(\mathbf{a}\times\mathbf{x})=2\,\mathbf{a}$$ The problem is, I keep getting ...
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1answer
18 views

Prove Tetrahedron Opposite Vectors add to $0$

I really need help on this problem, I'm in Multivariable Calculus (Calc III) and I just can't solve this. Let $v_1$, $v_2$, $v_3$, and $v_4$ be vectors whose lengths are equal to the areas of the ...
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1answer
22 views

Working with vectors in Linear Algebra

I'm just kinda confused about a problem in my linear algebra textbook. Maybe one of you geniuses on here can help me out. ...
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1answer
47 views

Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
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1answer
35 views

How to show that vectors form a basis [closed]

Show that the following vectors form a basis for $\mathbb{C}^n$ (or $\mathbb{R}^n$): $${\bf a}_1 =\!\begin{bmatrix} 1\\ 1\\ \vdots \\ 1\\ 1\\ \end{bmatrix}\!, \ {\bf a}_2 =\!\begin{bmatrix} 1\\ 1\\ ...
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3answers
71 views

Addition of Vectors?

How do you explain in general the addition of any two vectors geometrically without reference to any coordinate system?
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2answers
19 views

“best practice” for finding perpendicular vectors

I'm working through some examples in Strang (Ed. 3) and the solution manual. A couple questions in a row now have been "find vectors perpendicular to the given vector as well as perpendicular to each ...
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4answers
47 views

Sin(x) + Sin(y)

When you add sound waves you are basically adding sine and cosine of certain multiples of x. Is Sin(x) + Sin(y) ... + Sin(n) = Sin(x+y...+n)? Is the same true for summation of cosines? I am making a ...
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1answer
34 views

Vector Space Axioms

I am having trouble understanding vector space axioms seen here (http://imgur.com/qKhgAXu). As an example say we define our potential vector space to be the set of all pairs of real numbers of the ...
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1answer
48 views

Unit vector symbols/names

I am currently studying vectors and matrices in 3 dimensions, my book calls the unit vectors i j and k, however I have seen them being called in other ways, such as: x-hat, y-hat and z-hat; or simply ...
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0answers
17 views

Perpendicular vectors laying on planes, lines.

I'm working through a textbook example that I'm stumped to visualize properly. The first question says "the vectors perpendicular to v = (1,1,1) lie on a _____?" The answer is "all vectors ...
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1answer
21 views

Calculating a vector perpendicular to a unit vector

"find unit vector x in direction of w = (2,1,2)" then "find vector perpendicular to x" so I calculate x as w/length(w) = w/sqrt(2^2 + 1^2 + 2^2) = w/sqrt(9) = w/3 and a perpendicular vector of w = ...
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1answer
28 views

Silly vector-related question

It's probably a silly question but, I don't understand this : How can this be always true when $\vec{v}.d\vec{v} = v.dv.cos(\alpha) = v.dv \Leftrightarrow \alpha = [2\pi]$ It seems ...
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1answer
30 views

Gradient/Curl/Divergence of a Single Vector

This is for a homework problem, but I'm asking just a conceptual question. Is it possible to use the del operator on a standard vector? That's what I'm being asked to do, but I thought that was ...
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3answers
36 views

Consider 3 linearly independent, n-dimensional vectors. Adding a fourth under this circumstance is not linearly independent. Why?

Suppose that $\bf x$, $\bf y$, and $\bf z$ are three linearly independent, $n$-dimensional vectors. Define $\bf k$ by $$ {\bf k} = \alpha_1 {\bf x} + \alpha_2 {\bf y} + \alpha_3 {\bf z}, $$ where ...