Which graph represents vector 𝚨 added to vector 𝚩, where vector 𝚨 is equal to the
vector three, four and vector 𝚩 is the vector four, one, options (A), (B), (C),
(D), and (E)?
In this question, we’re given five options to determine which one correctly
represents the sum of two vectors, and we’re given the components of both of these
vectors 𝚨 and 𝚩. We can start by recalling exactly what is meant by this component notation for
vectors. The first component tells us the displacement of the vector in the horizontal
direction and the second component tells us the displacement of the vector in the
vertical direction. For example, vector 𝚨 has components three and four. The first component tells us its displacement in the horizontal direction. Positive three means we move three units to the right. The second component positive four tells us the displacement in the vertical
direction. We move four units up.
This means in the correct diagram vector 𝚨 should represent moving three units right
and four units up. We can see this is not true in option (A). In option (A), if we were to move along vector 𝚨, we would move three units to the
right. However, we would move five units up, so option (A) is not correct. Similarly in option (B), if we were to follow vector 𝚨, we see we move four units to
the right and three units up. So option (B) is also not correct. Similarly in option (C), we can see that vector 𝚨 is four units to the right and
three units up. So option (C) is also not correct.
Both options (D) and (E) correctly represent vector 𝚨. We move three units to the right and four units up. It’s also worth noting both of these two options correctly represent vector 𝚩. Vector 𝚩 has a positive displacement of four in the horizontal direction and a
positive displacement of one in the vertical direction. Vector 𝚩 represents moving four units right and one unit up. So now we can move on to determining vector 𝚨 plus vector 𝚩.
Remember, to add two vectors together, we want to add their displacements
together. And since we’re given the horizontal and vertical displacements of vectors 𝚨 and 𝚩,
we can add the two vectors together by adding their horizontal and vertical
displacements. Vector 𝚨 plus vector 𝚩 has horizontal component three plus four and vertical
component four plus one. And if we evaluate each of these components, we see it’s the vector seven, five. And we can see that only option (E) shows that the vector 𝚨 added to the vector 𝚩
represents the displacement of seven units right and five units up.
And it’s also worth noting we could’ve done this directly from the diagram by
sketching the vector 𝚩 to have its initial point at the terminal point of vector
𝚨. If we translate vector 𝚩 so that it starts at the terminal point of vector 𝚨, we
can see that it matches up with vector 𝚨 plus vector 𝚩, once again showing that
this correctly sums the two vectors, since the sum of these two vectors adds their
displacements. Therefore, we were able to show option (E) correctly represents vector 𝚨 added to