Which of the following sets of vectors is orthogonal?
$(a)v_{1}=(2,4,-2), v_{2}=(1,4)\\ (b)v_{1}=(14,5)^{T}, v_{2}=(10,-2)^{T}\\(c)v_{1=}(5,-2,3)^{T}, v_{2}=(-2,4,6)\\(d)v_{1=}(2,3)^{T}, v_{2}=(-1,4)^{T}$
Trail: I calculated the dot products and got results of $18,130,0,10$ simultaneously. So $(c)$ is the correct answer. Here I wonder can we calculate the dot product of 3 dim and 2 dim vectors as in $(a)$. similarly, is the calculation of dot product of row and column vector correct in case of $(c)$.
No, you cannot dot two vectors of different sizes, unless you pad one vector (or trim the other).
The second question is a bit tricky. Given that with two vectors $a \cdot b = a^\top b$ and $a \odot b= a b^\top$, the first being the inner product and the second being the outer product. So the dot product is always the matrix product of a row vector with a column vector. But I like the separate the meaning of the dot product from the details of how to calculate it. So I vote that the dot product does not care if the vectors are row or column vectors, but matrix products do care.