If two vectors $\vec{A} =A_x\hat{i} + A_y \hat{j} + A_z \hat{k}$ and  $\vec{B} =B_x\hat{i} + B_y \hat{j} + B_z \hat{k}$,  have angle $\theta$  between them then the dot product (scalar product) of $\vec{A}$ and $\vec{B}$ is  $$\vec{A}\cdot\vec{B} = |\vec{A}||\vec{B}|\cos \theta$$  $$\vec{A}\cdot\vec{B} =\left(A_x^2 +A_y^2+A_z^2\right)^{1/2}\left(B_x^2 +B_y^2 +B_z^2\right)^{1/2} \cos \theta $$  $$A_x B_x+ A_yB_y+A_zB_z=\left(A_x^2 +A_y^2+A_z^2\right)^{1/2}\left(B_x^2 +B_y^2 +B_z^2\right)^{1/2} \cos \theta  $$  Can we mathematically derive the last equation from right to left or left to right?  Is there any proof stating  that the equation to be practically and mathematically true?
24509
I have an intuitive understanding of why $a\dot{}b=|a||b|\cos{\theta}$ geometrically. The projection of one vector onto another makes sense to me when explaining the origin of this geometric definition.  What I don't understand is why $a\dot{}b=a_xb_x + a_yb_y = |a||b|\cos{\theta}$. How does the algebraic version of the dot product connect to the geometric version? Can you derive the algebraic definition from the geometric? I read the answers , but the proofs seem to depend on the actual algebraic definition to arrive at it.  My main question is, why are the two definitions really equal?
16281
Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product equation as a determinant of a carefully-constructed matrix,   but what I want to ask is... How did the dot and cross product come to be? When were they "invented"? Some detailed proofs? Did someone say: "Hey, wouldn't it be nice if we could construct a way to calculate a vector that is perpendicular to two given operands?"  Basically, how/why do they work?  I would appreciate explanations, links to other explanations, other web resources... I've been searching the Internet lately for explanations, but most of them are on how to use it and nothing that really gives substance to it.
68380
Let $A$ and $B$ be two constant matrices and let $x$ and $ y$ be two random vectors, what is the general formula for $Var(Ax+By)$? I know the formula for when $x$ and $y$ are scalar random variables and $A$ and $B$ are constants, but what about the matrix case?
16741
If $a$ and $b$ are vectors such that $\|a\| = 4$, $\|{b}\| = 5$, and $\|{a} + {b}\| = 7$, then find $\|2 {a} - 3 {b}\|$.  I couldn't figure out how to start off this problem. I attempted to use $$\cos \theta = \frac{a\cdot b}{\|a\|\cdot \|b\|}$$ But I still don't know what to do. Could someone nudge me in the right direction?  Thank you!
80235
Prove that if $e_1, \dots e_n$ is an orthonormal basis of $ V$ and $v_1, \dots , v_n$ are vectors in $ V$ such that $$\|e_j - v_j\| &lt; \frac{1}{\sqrt{n}}$$ for each j, then $v_1, \dots v_n$ is a basis of $V$.  I have fiddled around with this for a while now unfortunately nothing came close, so I'll just say I have no idea how to do this. Help :(?
119041
Suppose that $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging the first two rows of $A$. Prove that $\det(B) = -\det(A)$.  By general $n \times n$ matrix, I mean $$\left[\begin{matrix} a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\ a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} \end{matrix} \right]$$ I'm thinking of using the cofactor expansion of the 2nd row of the first matrix, yielding $$ \det(A) = c_{21} a_{21} + c_{22} a_{22} + \cdots+ c_{2n} a_{2n}  $$ which will be positive or negative depending on $n$ and the values of $a_{21}, \ldots,a_{2n}$, and using the same on the first row in matrix $B$ to get $c_{11} a_{21}$ etc. and then use the fact that the power of the $c$ subscripts will alter $\det(B)$ to be the opposite sign, but I'm not sure how to express all of this mathematically in a proof directly. Help please?
47710
I have seen a definition of orthogonality between two vectors something like this:  $$&lt;\vec{u}, \vec{v}&gt; = u^T \cdot v = 0$$  I am wondering what is the purpose of using a transpose of a vector (in this case and in general). I have also seen this in the formula to find the projection of a vector over another, but I have used just the normal vector instead of its transpose and everything seems to work.
58472
$$ [(\vec{a}+\vec{b})\times(\vec{b}+\vec{c})]\cdot(\vec{c}+\vec{a})=2\vec{c}\cdot(\vec{b}\times\vec{a})$$  I'm supposed to prove that this is true for all vectors $a,b,c$, but I keep getting $4(\vec{a}\times \vec{b})\cdot \vec{c}=0$, which is obviously not true.   Seems to me like there is a mistake in the problem, that it should be $\vec{a}\times\vec{b}$ instead of $\vec{b}\times\vec{a}$. Or I'm doing it wrong?  I'm using facts that $\vec{a}\times\vec{a}=0$ and $[\vec{a},\vec{b},\vec{c}]=[\vec{c},\vec{b},\vec{a}]=-[\vec{b},\vec{a},\vec{c}]$.
81134
My teacher told me that Vectors are quantities that behave like Displacements. Seen this way, the triangle law of vector addition simply means that to reach point C from point A, going from A to B &amp; then to C is equivalent to going from A to C directly.  But what is the meaning of a product of vectors? I cannot imagine how a product of displacements would look like in reality. Also, how do we know whether we need the scalar (dot) product or a vector (cross) product?
70619
I am convinced that the following identity is true: \begin{equation} \det\begin{bmatrix}  1+a_1^2 &amp; a_1 a_2 &amp; a_1 a_3 &amp; \ldots &amp; a_1a_n \\  a_1a_2 &amp; 1+a_2^2 &amp; a_2a_3 &amp; \ldots &amp; a_2a_n \\  \ldots &amp;\ldots &amp;\dots &amp; \ldots &amp; \ldots \\ a_na_1 &amp;a_na_2 &amp; a_na_3 &amp;\ldots &amp;1+a_n^2 \end{bmatrix}= 1+ a_1^2+a_2^2+\dots+a_n^2. \end{equation} Can you help me proving it?    This determinant comes out in the computation of the volume form on a n-dimensional surface on $\mathbb{R}^{n+1}$ described by the equation  \begin{equation} x_{n+1}=f(x_1\ldots x_n). \end{equation} The volume form is given by $\sqrt{g}dx_1\ldots dx_{n}$, where $g$ is the determinant above with $a_j=\partial_{x_j}f$. The result proven in this question &amp; answer shows that the volume form is  $$ \sqrt{1+\sum_{j=1}^n \left(\frac{\partial f}{\partial x_j}\right)^2}\, dx_1\ldots dx_n.$$
88846
i have to prove:   $$\begin{vmatrix} -2a &amp;a+b  &amp;a+c \\   b+a&amp;-2b  &amp;b+c \\   c+a&amp;c+b  &amp;-2c  \end{vmatrix} = 4(a+b)(b+c)(c+a)$$   I have tried many calculations between the rows and columns of the determinant to get to the answer i want to solve the exercise however none of them gave me something right. Can someone help?
96911
which vectors are perpendicular to each other?  $\vec a = (1, -2, 3)$, $\vec b = (5, 4, 1)$, $\vec c = (1, 0, -5)$  Do i just take the dot product of 2 of them. If the dot product they are at $90^\circ$? But how do i know if there perpendicular?
10958
I know by brutal calculation this identity holds always:  $$(u × v) \cdot (x × y) = \begin{vmatrix}  u \cdot x &amp; v \cdot x \\  u \cdot y &amp; v \cdot y \\  \end{vmatrix}$$  for arbitrary vectors $u$, $v$, $x$, $y$.  I'd like to know where it come from naturally.  Thank you.
91030
It's just a doubt.  $\vec { A } = A \hat{ x }  + A\hat { y} $  $ A\hat { y} =A\hat{ x} $  $\vec { A } = A \hat { x }  + A\hat { x} $  Now finding the magnitud of the vector,  $A=\frac { 1 }{ \sqrt { 2 }  } $  So the components of the unit vector are:  $\vec { A } = \frac { 1 }{ \sqrt { 2 }  } \hat { x } +\frac { 1 }{ \sqrt { 2 } } \hat { y} $  Is my answer right?
34807
I have faced difficulties while trying to prove that  $$\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$$   I don't have any clue how can I start to work with it. Any hint will be helpful.
5168
I understand the proof that  $$Var(aX+bY) = a^2Var(X) +b^2Var(Y) + 2abCov(X,Y), $$  but I don't understand how to prove the generalization to arbitrary linear combinations.  Let $a_i$ be scalars for $i\in {1,\dots ,n}$ so we have a vector $\underline a$, and $\underline X = X_i,\dots ,X_n$ be a vector of correlated random variables. Then $$ Var(a_1X_1 + \dots a_nX_n) = \sum_{i=1}^n a_i^2 \sigma_i^2 + 2 \sum_{i=1}^n \sum_{j&gt;i}^n a_i a_j \text{ Cov}(X_i,X_j)$$  How do we prove this? I imagine there are proofs in the summation notation and in vector notation?
22539
$\mathbf{a}\times \mathbf{b}$ 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 in the way that it is? I am mainly interested in the right hand rule defintion too as it is out of reach?