# A question on vector summing

by mech-eng
Tags: summing, vector
 P: 174 Hi, we can not sum a colon vector to a row vector, both with same number of elements. But, say A=[1 2 3] here first elements represents i, second j, and third k and say B=[1 3 5], here also first element represent i, second j, and third k. According to mathematical rules we can not sum them but we always sum them in physics. So I can not understand where I am wrong? (By B, I tried to write a column vector.)
 Homework Sci Advisor HW Helper Thanks P: 12,979 We do not sum column with row vectors in physics. Cartesian i-j-k vectors always have the same representation (either row or column, not both) - you have to pick a representation when you set up the geometry just like you have to pick a direction to call positive. Can you show me a reference where an authoritative text or a paper has summed row and column vectors?
P: 174
 Quote by Simon Bridge We do not sum column with row vectors in physics. Cartesian i-j-k vectors always have the same representation (either row or column, not both) - you have to pick a representation when you set up the geometry just like you have to pick a direction to call positive. Can you show me a reference where an authoritative text or a paper has summed row and column vectors?
I am asking the reason why we can not sum them. Let's say we want to sum A=1i +2j +3k
with B=2i + 3j + 5k . We can sum them easily by adding each element to each other but I ask
about If we write one of them as column and another as row why we can not sum them?
what is this reason? We just have to assume right of the equal sign row or column.

 HW Helper P: 1,808 A question on vector summing Because you can't sum matrices that have different shapes. It's not a defined operation.
P: 2,853
 Quote by mech-eng I am asking the reason why we can not sum them. Let's say we want to sum A=1i +2j +3k with B=2i + 3j + 5k . We can sum them easily by adding each element to each other but I ask about If we write one of them as column and another as row why we can not sum them? what is this reason? We just have to assume right of the equal sign row or column.
Because there is no utility in defining such a sum. It's much simpler to always represent a displacement vector as a row vector or a column vector, not both. You can define whatever you want in math, but unless it's useful, don't expect other people to follow you.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,569 You cannot add column and row vectors because they lie in different vector spaces! Given any vector space, V, its dual space, V*, is the set of all linear functions from V to the real numbers. It can be shown that, for V finite dimensional, the dual space is isomorphic to the original space using the isomorphism that maps the basis vector, $v_i$, to the to the linear function, $v_i^*$ that maps $v_i$ to 1 and maps all other basis vectors to 0. A way of representing that is to represent the vector $v= \sum a_iv_i$ as the "column vector" having $a_i$ as its "ith" component. We can then represent the function that maps the ith basis vector $e_i$ to 1 and all others to 0 as the row vector having "1" in its ith position. In that case, $f(v)$ is the matrix product of the row matrix representing f and the column matrix representing v.
P: 477
 Quote by mech-eng I am asking the reason why we can not sum them. Let's say we want to sum A=1i +2j +3k with B=2i + 3j + 5k . We can sum them easily by adding each element to each other but I ask about If we write one of them as column and another as row why we can not sum them? what is this reason? We just have to assume right of the equal sign row or column.
I don't see any reason why we couldn't define an "addition-like" operation between row and column vectors in the way which you seem to have in mind. The issues, which other posters have mentioned (or at least alluded to), are (1) whether this operation is useful and (2) whether this operation can properly be called a sum.

Considering the matrix product is defined on pairs of matrices with different shapes, and considering that this product is useful (though maybe not immediately so), and considering that we call it a product despite the fact that its inputs lie in different algebraic structures, I don't see any reason why your operation can't be called a sum and why it's necessarily not useful.
P: 446
 Quote by mech-eng I am asking the reason why we can not sum them. Let's say we want to sum A=1i +2j +3k with B=2i + 3j + 5k . We can sum them easily by adding each element to each other but I ask about If we write one of them as column and another as row why we can not sum them? what is this reason? We just have to assume right of the equal sign row or column.
Apples and oranges is the short answer. HallsofIvy has the long answer. The inbetween answer is what matterwave says, but I would add that if you have vectors

##A=\left[\begin{array}{1}a_1\\a_2\\a_3\end{array}\right]##

and

##B=\left[b_1\ b_2\ b_3\right]##

you can always add ##A^T+B## to get a row vector or ##A+B^T## to get a column vector. Why you would want to do that beats me, but if it ever happened, it is easy enough to do. What would ##A+B## equal? A row vector? A coumn vector? Neither makes sense. Sure, we could define ##A+B\equiv A^T+B##, but why?

Note also that one often defines ##i=[1\ 0\ 0]##, ##j=[0\ 1\ 0]##, and ##k=[0\ 0\ 1]## or the column versions.