Video Transcript
True or False: In the given figure,
vector 𝐂 is equal to vector 𝐀 plus vector 𝐁.
In this question, we’re given a
figure involving three vectors, 𝐀, 𝐁, and 𝐂. And we need to determine if vector
𝐂 is equal to the sum of vectors 𝐀 and 𝐁. To do this, let’s recall what we
mean by the sum of two vectors. First, a vector is an object
defined entirely in terms of its magnitude and direction, and we can often think of
this in terms of displacements. And we can represent this
graphically by using arrows. The length of the arrow tells us
the magnitude of the vector, and the direction the arrow points tells us the
direction of the vector. We can then think of the addition
of two vectors as combining their displacements. We just need to add the horizontal
and vertical displacements together. And there’s many different ways of
doing this. However, since we’re given a
figure, we’re going to go through one of the methods we can use by using a
figure.
Let’s start by looking entirely at
vector 𝐂 in our diagram. We can find the initial point of
vector 𝐂 by looking at the tail of the arrow, and we can find the terminal point of
vector 𝐂 by looking at the tip of the arrow. And we can think about this in
terms of displacements from the initial point of vector 𝐂 to its terminal
point. We can do the exact same thing with
vectors 𝐀 and 𝐁. And if we do this, we’ll notice
something interesting. Let’s do the exact same thing for
vector 𝐀. The initial point of vector 𝐀 is
given by the tail of its arrow, and the terminal point of vector 𝐀 is given by the
tip of its arrow. Similarly, for vector 𝐁, the
initial point of vector 𝐁 is given by the tail of its arrow, and the terminal point
of vector 𝐁 is given by the tip of its arrow.
And we can then notice something
interesting. If we start at the initial point of
vector 𝐀 and follow it all the way to its terminal point, we see that this is the
initial point of vector 𝐁. We can then follow the vector 𝐁
all the way to its terminal point to note that this is the terminal point of vector
𝐂. In other words, if we combine the
displacements of vectors 𝐀 and 𝐁, we get the exact same displacements as vector
𝐂. This is often called the
tip-to-tail method of adding two vectors together, since we draw the tip of vector
𝐀 to be the same as the tail of vector 𝐁. And this allows us to conclude that
the answer is true. If we add the displacements of
vectors 𝐀 and 𝐁 together, we get vector 𝐂. 𝐂 is equal to 𝐀 plus 𝐁.
