Video Transcript
Let ๐ณ be the zero vector. What is the vector ๐ณ plus the vector ๐ฎ equal to for any vector ๐ฎ?
In this question, weโre told that ๐ณ is the zero vector. And we need to find an expression for the vector ๐ณ plus the vector ๐ฎ for any vector ๐ฎ. And thereโs a few different ways we can do this. Letโs 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 ๐ฎ for any vector ๐ฎ. And it might be tempting to try and do this by adding the components together. However, itโs actually easier to do this graphically.
Recall, we can add two vectors ๐ฏ and ๐ฎ graphically. First, we sketch the vector ๐ฏ and then at the terminal point of vector ๐ฏ, we sketch the vector ๐ฎ. Then the vector ๐ฏ plus ๐ฎ will be the vector starting at the initial point of vector ๐ฏ and ending at the terminal point of vector ๐ฎ. 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 ๐ฎ. First, weโll want to draw our zero vector. But we know all components in the zero vector are equal to zero. So our zero vector doesnโt represent movement in any direction. So it doesnโt really matter how we draw our vector ๐ฎ. Since graphically, the zero vector doesnโt represent movement at all, moving along the zero vector and then moving along the vector ๐ฎ will always just be equal to the vector ๐ฎ. Therefore, weโve shown the zero vector plus the vector ๐ฎ will just be equal to the vector ๐ฎ.
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 ๐ฎ.
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 ๐ณ was the ๐-dimensional zero vector โ so thatโs just a vector of ๐ components, all of which are equal to zero โ and ๐ฎ is also a vector of ๐ dimensions โ weโll call the components ๐ฎ one and ๐ฎ two all the way up to ๐ฎ ๐ โ 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 ๐ณ are zero, so we just get the vector zero plus ๐ฎ one, zero plus ๐ฎ two, all the way up to zero plus ๐ฎ ๐. And we know that adding zero doesnโt change a value, so this just gives us the vector ๐ฎ one, ๐ฎ two, all the way up to ๐ฎ ๐, which was our vector ๐ฎ. Therefore, weโve shown for any vector ๐ฎ and zero vector of equal dimension, ๐ณ plus ๐ฎ will be equal to ๐ฎ and ๐ฎ plus ๐ณ will also be equal to ๐ฎ.
