Question Video: Checking a Possible Geometric Property of Vectors Mathematics

Video Transcript

True or False: In the given parallelogram, the vector 𝐂 is equal to the sum of vectors 𝚨 and 𝚩.

In this question, we’re given a parallelogram whose sides are represented by vectors 𝚨 and 𝚩, and the diagonal originating from vectors 𝚨 and 𝚩 is called vector negative 𝐂. We need to determine using this diagram if vector 𝐂 is equal to the sum of vectors 𝚨 and 𝚩. And since we’re given a diagram in which we’re told the shape is a parallelogram, let’s add these two vectors 𝚨 and 𝚩 by using the diagram. And we can do this by recalling the triangle rule for vector addition. This tells us we can add two vectors together given graphically if the terminal point of one vector is equal to the initial point of the other vector. We add the two vectors together by using the initial point of the first vector and the terminal point of the second vector.

In the given diagram, it’s worth noting vectors 𝚨 and 𝚩 currently have the same initial point; they’re not drawn tip to tail. However, we can find a vector equivalent to vector 𝚩 which starts at the terminal point of vector 𝚨. We can do this by noting the given shape is a parallelogram, and opposite sides in a parallelogram are parallel and have the same length. This means if we represent the side opposite vector 𝚩 as a vector, it has the same length, which means it has the same magnitude and the sides are parallel. So these vectors will have the same direction. In other words, opposite sides in a parallelogram are represented by the same vector. Both of these sides are vector 𝚩.

And now that the terminal point of vector 𝚨 and the initial point of vector 𝚩 are coincident, we can add these two vectors together by using the triangle rule for vector addition. We get this diagonal is equal to vector 𝚨 plus vector 𝚩. In other words, 𝚨 plus 𝚩 is equal to negative 𝐂. However, we’re asked in the question if this is equal to 𝐂, and of course negative 𝐂 is not equal to 𝐂. Therefore, we can conclude that the answer to this question is false. In the given parallelogram, vector 𝚨 plus vector 𝚩 is equal to vector negative 𝐂, not vector 𝐂.