Let 𝐳 be the zero vector. What is the vector 𝐳 plus the vector 𝐮 equal to for any vector 𝐮?
In this question, we’re told that 𝐳 is the zero vector. And we need to find an expression for the vector 𝐳 plus the vector 𝐮 for any vector 𝐮. And there’s a few different ways we can do this. Let’s start by recalling exactly what we mean by the zero vector.
The zero vector is the vector which has all of its components equal to zero. So for example, the vector zero, zero will be the zero vector in two dimensions. We need to find an expression for the zero vector plus the vector 𝐮 for any vector 𝐮. And it might be tempting to try and do this by adding the components together. However, it’s actually easier to do this graphically.
Recall, we can add two vectors 𝐯 and 𝐮 graphically. First, we sketch the vector 𝐯 and then at the terminal point of vector 𝐯, we sketch the vector 𝐮. Then the vector 𝐯 plus 𝐮 will be the vector starting at the initial point of vector 𝐯 and ending at the terminal point of vector 𝐮. And although in this case our sketch is in two dimensions, this is also true in any number of dimensions.
We can use this to find an expression for the zero vector plus the vector 𝐮. First, we’ll want to draw our zero vector. But we know all components in the zero vector are equal to zero. So our zero vector doesn’t represent movement in any direction. So it doesn’t really matter how we draw our vector 𝐮. Since graphically, the zero vector doesn’t represent movement at all, moving along the zero vector and then moving along the vector 𝐮 will always just be equal to the vector 𝐮. Therefore, we’ve shown the zero vector plus the vector 𝐮 will just be equal to the vector 𝐮.
And we can also point out something else here. We know that vector addition is commutative, so we could actually add these in the opposite order. We would still get the vector 𝐮.
And there is one more way we could have proven this. We could have proven this directly from component addition of two vectors. So for example, if 𝐳 was the 𝑛-dimensional zero vector — so that’s just a vector of 𝑛 components, all of which are equal to zero — and 𝐮 is also a vector of 𝑛 dimensions — we’ll call the components 𝐮 one and 𝐮 two all the way up to 𝐮 𝑛 — then we know how to add these two vectors together. We do this component-wise. We just add the corresponding components of these two vectors together.
But all of the components of 𝐳 are zero, so we just get the vector zero plus 𝐮 one, zero plus 𝐮 two, all the way up to zero plus 𝐮 𝑛. And we know that adding zero doesn’t change a value, so this just gives us the vector 𝐮 one, 𝐮 two, all the way up to 𝐮 𝑛, which was our vector 𝐮. Therefore, we’ve shown for any vector 𝐮 and zero vector of equal dimension, 𝐳 plus 𝐮 will be equal to 𝐮 and 𝐮 plus 𝐳 will also be equal to 𝐮.