Video: Identifying Vectors Equivalent to the Sum of Vectors in Geometric Contexts

The figure shows a regular hexagon 𝐴𝐡𝐢𝐷𝐸𝐹 divided into 6 equilateral triangles. Which of the following is equal to 𝐁𝐄 + 𝐅𝐀? [A] 𝐃𝐂 [B] 𝐁𝐀 [C] 𝐅𝐃 [D] 𝐂𝐃 [E] 𝐄𝐅

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Video Transcript

The figure shows a regular hexagon 𝐴𝐡𝐢𝐷𝐸𝐹 divided into six equilateral triangles. Which of the following is equal to 𝐁𝐄 plus 𝐅𝐀? (A) 𝐃𝐂, (B) 𝐁𝐀, (C) 𝐅𝐃, (D) 𝐂𝐃, (E) 𝐄𝐅.

Looking at our figure, we see this regular hexagon and the six equilateral triangles that make it up. We can recall that an equilateral triangle is one where all the side lengths are the same. That means, for example, that this side length, 𝐹𝐴, is the same as this length or this length or any such similar triangle side in this hexagon. In our question statement, we’re told about two vectors: vector 𝐁𝐄 and vector 𝐅𝐀. On our figure, 𝐁𝐄 looks like this, a vector from point 𝐡 to point 𝐸. 𝐅𝐀 is like this. It’s half the length of 𝐁𝐄, and note that it points in the opposite direction.

That part about direction is important because it means when we go to add 𝐁𝐄 and 𝐅𝐀, the result will look like this, a vector with the same length as 𝐅𝐀, but it points in the same direction as 𝐁𝐄. Note that to find this result, we’ve effectively used the tip-to-tail method of vector addition. That is, we’ve taken vector 𝐁𝐄 then put the tail of vector 𝐅𝐀 at the tip of 𝐁𝐄. And this shows us that the resultant of these two vectors, their vector sum, goes from the tail of 𝐁𝐄 to the tip of 𝐅𝐀. This means that among our five answer options, we’re looking for the vector that looks like this one on our diagram. It will point straight upward like vector 𝐁𝐄, and it will have the same length or magnitude as vector 𝐅𝐀.

If we first look at option (A), vector 𝐃𝐂, that would look like this on our figure. It has the right magnitude but the wrong direction. Moving on to vector 𝐁𝐀, this vector also has the correct length or magnitude but the incorrect direction. Vector 𝐅𝐃 in option (C) looks like this. This option is correct neither in direction nor magnitude. But then, if we look at vector 𝐂𝐃, this is the vector that goes from point 𝐢 to point 𝐷. We see this has the same length as 𝐅𝐀, like we wanted, but it points upward like vector 𝐁𝐄. Vector 𝐂𝐃 then is a match for our resultant vector. For completeness sake, let’s look at the last option vector 𝐄𝐅. This vector also has the correct magnitude but the incorrect direction.

Of these answer options then, we find that it’s vector 𝐂𝐃 that’s equal to 𝐁𝐄 plus 𝐅𝐀.

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