1-D vectors and scalars

Question

So we had a debate (more like argument) over comparing a vector to a scalar. We agreed you can't, but couldn't come to an agreement over whether you can if it's in 1D.

Consider A as a 1D vector and B a scalar. Is it completely incorrect to say A=B. Usually I'd say yes, a vector can never equal a scalar. But in 1D isn't the idea kind of interchangeable in certain cases.

Dot product with scalar equal to common symbol for multiplication

Gave some insight, but I'd like a definite yes or no answer.

Can you have A=B where A is a 1D vector and B is a scalar value in any scenario. I'm not worried about in general. I'm just curious if it is possible in at least one case, and if so, which case is this.

Note, it's 'basically' as if we are saying a scalar is a 1D vector. That's the only way they can be equal. — Kendall, Aug 02 '17 at 16:29
When the vector space changes so do the vectors. What I mean is, $a\in \Bbb R$ is a vector in the space $\Bbb R$, but not a vector in $\Bbb R^2$ — Naive, Aug 02 '17 at 16:31

score 0 · Accepted Answer · answered Aug 02 '17 at 17:00

At the informal level, when we talk about a scalar, it is a quantity with magnitude but no direction, and when we talk about a vector, it is a quantity with magnitude and direction, so we never compare a scalar with a vector. Here we imagine a vector as a directed line segment as they are represented in say high school textbooks.

At the formal level, the set of all real numbers $\mathbb R$ is a one-dimensional real vector space, and its elements are vectors. Obviously, these real numbers are also scalars. Here the scalar and the vector are indeed the same mathematical object.

1-D vectors and scalars

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