Why is $a\hat{\imath} + b\hat{\jmath } + c\hat{k}$ meaningful when $\hat{\imath}$, $\hat{\jmath }$, $\hat{k}$ are not 'alike' quantities? For the standard form: $a\hat{\imath} + b\hat{\jmath } + c\hat{k}$.
Since the $\hat{\imath}$, $\hat{\jmath }$, and $\hat{k}$ directions are different, why are we 'allowed' to write them this way? Isn't addition only allowed between 'alike' quantities, i.e. values along the same direction?
 A: If I need $2$ apples and $3$ oranges for some recipe I want to prepare, it would be perfectly fine for me to write
$$
  2~\text{apples} + 3~\text{oranges}
$$
on my shopping list. However, it would not be fine to write
$$
  5~\text{apples and/or oranges}
$$
because I might end up with $4$ apples and $1$ orange, unable to prepare my meal.
So when we say that you can only add “alike” quantities, we mean that you can only simplify “$2$ of something plus $3$ of something” to “$5$ of something” if all the somethings are the same.
The same applies to your vectors:
$$
  a\hat{\imath} + b\hat{\jmath } + c\hat{k}
$$
is fine, but we can’t simplify this to
$$
  (a + b + c) \hat{?}
$$
(I don’t even know what you might use for $\hat{?}$). In this case, we do introduce new notation, though, and also allow writing the result as $(a, b, c)$. This is fine because we can still extract each component from this result.
A: You can think $v=a \hat \imath+b \hat \jmath +c \hat k$ as walking $a$ steps in direction $\hat \imath$, after walking $b$ steps in direction $\hat \jmath $ and finally walking $c$ steps in direction $\hat k$. After these three walks you end up in a position of the space, and the segment from the origin of your movement to your actual position defines geometrically the vector $v$.
As $a, b,c$ are not generally integers then, instead of steps you can think that you travel distances $a,b$ or $c$ in the corresponding directions $\hat \imath, \hat \jmath $ or $\hat k$.
It can be shown that in the space we can use these three basic directions $\hat \imath,\hat \jmath ,\hat k$ to define any position on the space from the origin, so it defines any vector.
A: In vector space you can add vectors and multiply them by a scalar to get a new vector:
$$
\vec w =  \alpha\vec u+\beta\vec b.
$$
It is the main property of a vector space.
Since three vectors $\hat\imath$, $\hat\jmath$ and $\hat k$ belong to the same vector space, you can do an algebraic sum with them.
A: Just think of decomposing the space into three perpendicular directions, where each direction {i,j,k} is "independent" of each other. Here the plus sign is not real plus of quantities, it is plus of vectors. i,j,k are all vectors.
$\hat\imath$ = [1,0,0]
$\hat\jmath$ = [0,1,0]
$\hat k$ = [0,0,1]
A: The principal reason is because in a finite $n$ dimensional vector space $V$ over the field $F$ there are an object called the basis of the vector space and usually is denoted by $B=\lbrace v_1,v_2,\ldots, v_n\rbrace$ which is a special set with the property that any vector $v\in V$ can be written uniquely as linear combination of elements of $B$, it is for any $v\in V$, we can write $$v=\sum_{j=1}^n \alpha_i v_i$$ for $\alpha\in F$ and $F$ a field.
In particular $\mathbb{R}^3$ is a vector space with basis $B=\lbrace e_1,e_2,e_3\rbrace$ called the canonical basis, where $e_i$ is the vector of all the entries $0$ except the $i$ coordinate which have a $1$.
And as I tell in a more abstract sense any vector $v$ can be written as $$v=\alpha_1e_1+\alpha_2e_2+\alpha_3e_3$$ but in engineering texts the vectors $e_1,e_2,e_3$ are denoted usually by $i,j,k$
A: Not sure of the exact history, but to my knowledge the convention can be traced to the origins of the complex numbers $\mathbb{C}$. Folk in the $17$th century were after solutions to cubic equations, which lead them to the mysterious number $i$, which crucially is $2$-dimensional, but appears $1$-dimensional, in the sense that it lies on the $y$-axis (of course, understanding this clears up any mystery surrounding $i$). Since $i$ proved spectacularly successful in solving cubics, folk then took the natural step to generalise the algebraic structure $\mathbb{R}$ (a field in modern parlance) to a wider setting of complex numbers called $\mathbb{C}$, with the familiar notation $$z=x+iy$$ and rules for adding and multiplying the new numbers, which ensure $\mathbb{C}$ is "closed" under these operations. This means, for any $z,u\in\mathbb{C}$, $$z+u\in\mathbb{C}, \qquad zu\in\mathbb{C}.$$
Notice in this form, complex numbers still feel $1$-dimensional. William Hamilton was one of the first to succinctly state the geometric significance of complex arithmetic, as the addition and multiplication of ordered pairs of numbers. Hamilton saw clearly that $\mathbb{C}$ is an algebra of $2D$ numbers, or vectors. With this, he set out to discover an equivalent algebra of $\mathbb{R}^3$. It took about a decade before he realised that this was impossible! But he made the marvelous discovery that, if you give up the dream of multiplicative commutativity, you can define a normed division algebra for $\mathbb{R}^4$, which he called the quaternions $\mathbb{H}$. This idea was so bright Hamilton famously carved its germ onto a bridge when inspiration flashed upon him during a walk. The image he carved in stone was an equation $$\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1.\tag{*}$$ A quaternion is usually written in the form $$\mathbf{q}=a\mathbf{1} +b\mathbf{i}+c\mathbf{j}+d\mathbf{k},$$ where $\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}$ are  $2\times2$ complex matrices. Hence quaternion algebra is matrix alebgra. Since $\mathbf{q}$ is uniquely determined by four real numbers $a,b,c,d$, we also have $\mathbf{q}\in\mathbb{R}^4$. From this perspective $\mathbb{R}^3$ can be viewed as the subset of pure imaginary quaternions $$a\mathbf{i}+b\mathbf{j}+c\mathbf{k}.$$ Vector arithmetic eventually won out as the standard tool for $\mathbb{R}^3$, but it would seem some of Hamilton's notation wormed it's way into the theory.
Looking back, we can even see that the notation for complex numbers $z=x+iy$ is an elegant form of matrix algebra $$z=x+iy=x\mathbf{1}+y\mathbf{i}=x\begin{bmatrix}1 & 0\\0 &1\end{bmatrix}+y\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}=\begin{bmatrix}x & -y\\y & x\end{bmatrix}.$$  So, if you really want, you can view what you call the standard form as addition between like quantities as follows $$a\mathbf{i}+b\mathbf{j}+c\mathbf{k}=a\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}+b\begin{bmatrix}0 & -i\\-i & 0\end{bmatrix}+c\begin{bmatrix}i & 0\\0& -i\end{bmatrix}=\begin{bmatrix}ic & -a-ib\\a-ib & -ic\end{bmatrix}.\tag{**}$$ As a justification for why we might want to do this, it can be shown that, for any unit quaternion $\mathbf{t}$, the conjugation map $$\mathbf{q}\mapsto \mathbf{t}^{-1}\mathbf{q}\mathbf{t}$$ is a rotation of $\mathbb{R}^3$ when the $\mathbf{q}$ are taken as pure imaginary quaterions. For details check out On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by Conway and Smith, or Naive Lie Theory by Stillwell.
A: In 3D space, every point can be reached by a linear combination of three unit vectors: one in the $\hat{i}$ direction, one in the $\hat{j}$ direction, and one in the $\hat{k}$ direction.
$a\hat{i}$, $b\hat{j}$, and $c\hat{k}$ are all vectors (in this case, all $3$-dimensional vectors), so you can add them as if you were adding any other vectors like $(1,2)$ and $(4,7)$.
A: We are "allowed" to write vectors this way because, even we we pretend to forget that $\hat{i}$, $\hat{j}$, and $\hat{k}$ are not ordinary numbers, we still get the right answers when doing calculations. We only have to follow the rule that $\hat{i} + \hat{j}$, $\hat{i} + \hat{k}$, and $\hat{j}+ \hat{k}$ cannot be simplified.
Some examples:
Addition: $(3\hat{i} +  2\hat{j} - 4\hat{i}) + (5\hat{i} - \hat{j} + 2\hat{k}) = (3 + 5)\hat{i} + (2-1)\hat{j} + (-4 + 2)\hat{k} = 8\hat{i} + \hat{j} + -2\hat{k}$
This works because vector addition is associative, commutative, and distributes with real numbers. We can use strategies learned years earlier like combining like terms to get the final answer.
Dot product: $(3\hat{i} +  2\hat{j}) \cdot (5\hat{i} - \hat{j}) = (3)(5)\hat{i}\cdot\hat{i} + (3)(-1)\hat{i}\cdot\hat{j} + (2)(5)\hat{j}\cdot\hat{i} + (2)(-1)\hat{j}\cdot\hat{j} = (3)(5) + (2)(-1) = 13$
Here, we use the fact that $\hat{i}\cdot\hat{i} = \hat{j}\cdot\hat{j} = 1$ and $\hat{i}\cdot\hat{j} = 0$ to get our final answer. Otherwise, it's just the distributive property (the FOIL method in many schools).
I could do another example with the cross product, but that would be a lot of intermediate symbols coming from multiplying two three-term vector expressions. But, if you treat the cross product like ordinary multiplication and then simplify using the cross products of unit vectors, it would work. (This is called "leave it as an exercise to the reader.")
The purpose of any notation is to make concepts easier to think about, usually by reducing the number of rules we have to remember when doing calculations. Writing vectors as sums of simple vectors allows us to treat them like normal numbers and use the calculation rules that are much more familiar and ingrained in our minds. We only need to add a couple of rules for simple examples of the new kinds of quantities to finish the calculations.
This is similar to complex numbers. Writing them as $a + bi$ makes it very easy to remember how to multiply them using FOIL/distributive methods learned much earlier in school. You don't need a new multiplication rule to figure out $(a + bi)(c + di)$, only that $i^2 = -1$. If complex numbers were only written as ordered pairs, then you would have to memorize the seemingly arbitrary rule: $(a,b)\times(c,d)=(ac-bd,bc+ad)$. Even here, I used the $(a + bi)(c + di)$ notation to figure out the ordered pair notation because it's easier to think about, which is the purpose of notation.
