Does the vector sum the vector one, two added to the vector two, three, one have a solution?
In this question, we’re given a vector sum involving two vectors. We need to determine if we can add these two vectors together. To answer this question, we need to notice something interesting about the two vectors. The first vector is two-dimensional; it has two components. And the second vector is three-dimensional; it has three components. Let’s now recall the definition of vector addition. This tells us if we have two vectors 𝑢 and 𝑣, which are of equal dimension — say, 𝑢 is equal to the vector 𝑢 sub one, 𝑢 sub two and it has components all the way up to 𝑢 sub 𝑛, and 𝑣 is the vector 𝑣 sub one, 𝑣 sub two with components up to 𝑣 sub 𝑛 — then we can add vectors 𝑢 and 𝑣 together by adding the corresponding components together.
𝑢 plus 𝑣 is the vector 𝑢 sub one plus 𝑣 sub one, 𝑢 sub two plus 𝑣 sub two. And we keep adding components together of this form all the way up to 𝑢 sub 𝑛 plus 𝑣 sub 𝑛. And we can immediately see where the problem lies. To add two vectors together by adding the corresponding components together, each component must have a corresponding component. They must have equal dimension. However, the two vectors we’re given do not have equal dimension. The first vector has two components, so its dimension is two. And the second vector has dimension three; it has three components. Therefore, using this definition of vector addition, we cannot add the two vectors together.
The answer is no. The sum vector one, two added to vector two, three, one does not have a solution.