Video Transcript
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.
