I would like to know whether I am correct in my thinking below.

You can add two vectors using basic arithmetic only when either two cases occur:

(1) the vectors are parallel to each other
(2) the vectors direction are opposite of each other (in which case, you could multiply one of the vectors by -1 in order to make them be parallel to each other, and then (1) would apply).

Otherwise, you would have to use those fancy triangle laws and follow tip-to-tail methods etc.

2. ## Re: Vector addition cases

I am not sure what you mean.

$$\begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} + \begin{bmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix} = \begin{bmatrix}x_1+y_1 \\ x_2+y_2 \\ \vdots \\ x_n+y_n\end{bmatrix}$$

This has the effect of the tip-to-tail method. The resultant vector will be the third side of the triangle.

3. ## Re: Vector addition cases

Originally Posted by otownsend
I would like to know whether I am correct in my thinking below.
You can add two vectors using basic arithmetic only when either two cases occur:
(1) the vectors are parallel to each other
(2) the vectors direction are opposite of each other (in which case, you could multiply one of the vectors by -1 in order to make them be parallel to each other, and then (1) would apply).
I too am not sure what you mean. You seem to have a somewhat odd idea of vectors.
A vector is a hybrid for mathematics in that vectors have length & direction. Therefore, vectors are equivalent classes. Were use vector spaces to as models. You posted this in a calculus forum. So it is usual that we use $\mathbb{R}^n,~n=2,3,4$ as the setting for your problems.

Therefore, in a given space, say $\mathbb{R}^3$, we can add any two vectors. There are no restrictions as long as we are in the same space. (i.e. vectors from $\mathbb{R}^3$ cannot be added to vectors in $\mathbb{R}^n$ if $n\ne 3)$. Two vectors are parallel in the same direction if they are positive multiples of each other, if negative multiples then parallel in opposite directions.

4. ## Re: Vector addition cases

Perhaps you are meaning that you can add vector a with vector ka which is parallel to it to get a+ka = (1+k)a

but you can't algebraically simplify a+b if a and b are not parallel (ie not scalar multiples).

(using bold to represent vectors)

5. ## Re: Vector addition cases

Debsta's response was exactly what I was looking for. Thank you for your clarification!

6. ## Re: Vector addition cases

Originally Posted by otownsend
You can add two vectors using basic arithmetic only when either two cases occur:
(1) the vectors are parallel to each other
(2) the vectors direction are opposite of each other (in which case, you could multiply one of the vectors by -1 in order to make them be parallel to each other, and then (1) would apply).
Originally Posted by Debsta
Perhaps you are meaning that you can add vector a with vector ka which is parallel to it to get a+ka = (1+k)a
but you can't algebraically simplify a+b if a and b are not parallel (ie not scalar multiples).
(using bold to represent vectors)
Originally Posted by otownsend
Debsta's response was exactly what I was looking for.
It may have been what you were looking for, but it is not what you asked.
We can add any two vectors in the same space using simple number operations.
You seem to need to go to a good textbook on vector-calculus. Look into scalar multiplication.