Let a and b be $3D$ vectors. Which of the following expressions make sense? Let $a$ and $b$ be $3D$ vectors. Which of the following expressions make sense?
$A.
(a\cdot b)+a$
$ B.
(a\times b)+a$
$ C.
(a\times b)\cdot a$
$ D.
(a\times b)\times a$
 A: A little vector intuition:
A vector is a quantity that has direction and magnitude. In other words, it points somewhere and has a certain length. Vectors in 3d $(\mathbb{R}^3)$ are represented by an $x$, $y$, and $z$ coordinate, usually like follows: $\big<x, y, z\big>$.
There are two ways to multiply two vectors together. One way is the dot product, $a\cdot b$. This operation tells you to multiply the identical components of the two vectors together, and sum them. So, if you have these two vectors:
$$a=\big<1, 3, 0\big>\\b=\big<4, -1, 2\big>$$
$$a\cdot b=(1)(4)+(3)(-1)+(0)(2)=1$$
The dot product always gives you a scalar value, which is just a number (no direction).
The other way of multiplying vectors is by taking the cross product $(a\times b)$. You do this by taking the three by three determinant of the two vectors. I'm not going to explain how in this answer, because it'd take a while, but you can go here for a good explanation.
The point is, $a\times b$ gives you a vector (of form $(x, y, z)$), not a scalar.
So now to your question:
You cannot combine different types of values; imagine trying to add a number to a point $(x, y)$ - i.e. what is $(3, 2) + 4$? You can't do it. It's the same here. You cannot add scalars and vectors.
So in letter A, $a\cdot b$ gives you a scalar; $a$ itself is a vector. You cannot add scalars and vectors, therefore this is not really valid.
A: Follow these rules:


*

*(scalar)(vector) = (vector)

*(vector)+(vector) = (vector)

*(vector)·(vector) = (scalar)

*(vector)×(vector) = (vector)
and (scalar)+(vector) is invalid.
