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Question Video: Determining a Vector Graphically Mathematics

Which of the following is equivalent to 𝐀𝐁 βˆ’ 𝐃𝐁? [A] 𝐃𝐀 [B] 𝐃𝐂 [C] 𝐁𝐀 [D] 𝐁𝐂 [E] 𝐀𝐂

02:17

Video Transcript

Which of the following is equivalent to the vector from 𝐴 to 𝐡 minus the vector from 𝐷 to 𝐡? Is it option (A) the vector from 𝐷 to 𝐴? Option (B) the vector from 𝐷 to 𝐢. Option (C) the vector from 𝐡 to 𝐴. Is it option (D) the vector from 𝐡 to 𝐢? Or is it option (E) the vector from 𝐴 to 𝐢?

In this question, we’re given a quadrilateral 𝐴𝐡𝐢𝐷. And we can see in this quadrilateral the opposite sides are parallel. So 𝐴𝐡𝐢𝐷 is a parallelogram. And we need to use this diagram to determine which of five given vectors is equivalent to the vector from 𝐴 to 𝐡 minus the vector from 𝐷 to 𝐡. And there’s many different ways of answering this question.

The first thing we want to note is opposite sides in a parallelogram have the same length. So the side 𝐴𝐡 has the same length as the side 𝐷𝐢. And the side 𝐴𝐷 has the same length as the side 𝐡𝐢. Next, there’s a few different ways we can evaluate the vector subtraction. We’re going to do this graphically because we’re given a diagram. We’ll start by adding the vector from 𝐴 to 𝐡 onto our diagram. Its initial point is 𝐴, and its terminal point is 𝐡.

Next, instead of adding the vector from 𝐷 to 𝐡 on our diagram, we can note we’re subtracting this vector. And subtracting a vector between two points is the same as adding the vector where we switch its initial and terminal points. So negative the vector from 𝐷 to 𝐡 is equal to the vector from 𝐡 to 𝐷. We can then add the vector from 𝐡 to 𝐷 onto our diagram. Its initial point is 𝐡, and its terminal point is 𝐷.

And now we can add these two vectors together by using the triangle rule for vectors. Our initial point will be 𝐴, and our terminal point will be 𝐷. So we have the vector from 𝐴 to 𝐡 added to the vector from 𝐡 to 𝐷 is the vector from 𝐴 to 𝐷. And we could stop here. However, we can see the vector from 𝐴 to 𝐷 is not one of our five given options. So we’re going to need to find a vector which is equal to the vector from 𝐴 to 𝐷. And we do this by recalling for vectors to be equal, they need to have the same magnitude and direction. And we can see this by using our diagram. The vector from 𝐡 to 𝐢 is parallel to the vector from 𝐴 to 𝐷. And since this is a parallelogram, we know they have the same length. So the vectors have the same magnitude.

So the vector from 𝐴 to 𝐡 minus the vector from 𝐷 to 𝐡 is equal to the vector from 𝐡 to 𝐢, which we can see is option (D).

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