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

The figure shows a regular hexagon π΄π΅πΆπ·πΈπΉ divided into 6 equilateral triangles. Which of the following is equal to ππ + ππ? [A] ππ [B] ππ [C] ππ [D] ππ [E] ππ

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### Video Transcript

The figure shows a regular hexagon π΄π΅πΆπ·πΈπΉ divided into six equilateral triangles. Which of the following is equal to ππ plus ππ? (A) ππ, (B) ππ, (C) ππ, (D) ππ, (E) ππ.

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, πΉπ΄, is the same as this length or this length or any such similar triangle side in this hexagon. In our question statement, weβre told about two vectors: vector ππ and vector ππ. On our figure, ππ looks like this, a vector from point π΅ to point πΈ. ππ is like this. Itβs half the length of ππ, and note that it points in the opposite direction.

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

If we first look at option (A), vector ππ, that would look like this on our figure. It has the right magnitude but the wrong direction. Moving on to vector ππ, this vector also has the correct length or magnitude but the incorrect direction. Vector ππ in option (C) looks like this. This option is correct neither in direction nor magnitude. But then, if we look at vector ππ, this is the vector that goes from point πΆ to point π·. We see this has the same length as ππ, like we wanted, but it points upward like vector ππ. Vector ππ then is a match for our resultant vector. For completeness sake, letβs look at the last option vector ππ. This vector also has the correct magnitude but the incorrect direction.

Of these answer options then, we find that itβs vector ππ thatβs equal to ππ plus ππ.