Video Transcript
True or False: In the given
parallelogram, the vector 𝐂 is equal to the sum of vectors 𝚨 and 𝚩 is equal to
the sum of vectors 𝚩 and 𝚨.
In this question, we’re given a
parallelogram where the sides of the parallelogram are represented by two vectors,
𝚨 and 𝚩. We need to use this figure to
determine whether the diagonal of this parallelogram is represented by vector 𝚨
plus vector 𝚩 and vector 𝚩 plus vector 𝚨. And there’s a few different ways of
doing this. For example, we can already note
that vector 𝚨 plus vector 𝚩 is equal to vector 𝚩 plus vector 𝚨. Because this is called the
communicative property of vector addition. This is true for any two vectors,
so it must be true for vectors 𝚨 and 𝚩. We can add vectors in any
order.
So, we need to determine whether
vector 𝐂 is equal to the sum of these two vectors. And there’s a few different ways of
doing this. We’re going to use the triangle
rule for vector addition. And we can recall the triangle rule
for vector addition tells us if we have any three points 𝑃, 𝑄, and 𝑅, the vector
from 𝑃 to 𝑄 added to the vector from 𝑄 to 𝑅 is equal to the vector from 𝑃 to
𝑅. In other words, if we’re adding two
vectors together where the initial point of one vector is coincident with the
terminal point of the other vector, then we just take the vector with initial point
of the first vector and terminal point of the second vector.
And we can see this is exactly what
we have in the diagram. For example, we can see the
terminal point of vector 𝚨 is the point lowercase 𝑏. And this is the initial point of
vector 𝚩. Therefore, vector 𝚨 is the vector
from 𝑎 to 𝑏, and vector 𝚩 is the vector from 𝑏 to 𝑐. So, the sum of these two vectors is
equal to the sum of the vector from 𝑎 to 𝑏 and the vector from 𝑏 to 𝑐. And we can then apply the triangle
rule for vector addition to add these two vectors. We get the sum of vectors 𝚨 and 𝚩
is the vector from lowercase 𝑎 to lowercase 𝑐, which we can see is the given
diagonal in the figure.
And although this is technically
enough to answer our question, it’s worth noting we could have also done this by
using the other two sides in the parallelogram. We can note that vector 𝚩 has the
terminal point lowercase 𝑑, and lowercase 𝑑 is the initial point of vector 𝚨. So, we can use this to rewrite the
expression vector 𝚩 plus vector 𝚨. It’s equal to the vector from 𝑎 to
𝑑 added to the vector from 𝑑 to 𝑐.
And once again, we can apply the
triangle rule for vector addition to add these two vectors together. It’s also equal to the vector from
lowercase 𝑎 to lowercase 𝑐, which is the given diagonal in the figure. And the reason it’s important to
note that we can use either of the two sides given is that this is part of a proof
to show that vector addition is commutative.
We’ve shown if vectors 𝚨 and 𝚩
represent the sides of a parallelogram, then we can add the two vectors in either
order by using this method. And this means we were able to show
that it’s true in the given parallelogram, vector 𝐂 is equal to the sum of vector
𝚨 and vector 𝚩. And it’s also equal to the sum of
vector 𝚩 and vector 𝚨.
