Consider the given figure. If 𝐴𝐵𝐶𝐷 is a rectangle and 𝐸 is the midpoint of the line segment 𝐴𝐷, then vector 𝐄𝐁 plus vector 𝐁𝐀 minus vector 𝐃𝐂 is equal to blank. Is it (A) vector 𝐁𝐄, (B) vector 𝐂𝐄, option (C) vector 𝐄𝐁, or option (D) vector 𝐄𝐂?
We begin by recalling that two vectors are equal if they have the same direction and magnitude. This means that in a rectangle, the parallel sides will be represented by the same vectors as they have the same magnitude and direction. This means that vector 𝐁𝐀 is equal to vector 𝐂𝐃 and vector 𝐀𝐃 is equal to vector 𝐁𝐂. We are also told that 𝐸 is the midpoint of the line segment 𝐴𝐷. This means that vector 𝐀𝐄 is equal to vector 𝐄𝐃, and both of these are equal to one-half multiplied by vector 𝐀𝐃.
Let’s now consider the expression we are given, vector 𝐄𝐁 plus vector 𝐁𝐀 minus vector 𝐃𝐂. We know that the general vectors 𝐗𝐘 and 𝐘𝐗 are the additive inverse of each other. This means that vector 𝐗𝐘 plus vector 𝐘𝐗 is equal to the zero vector. It also means that the vector 𝐗𝐘 is equal to the negative of vector 𝐘𝐗. This means that instead of subtracting vector 𝐃𝐂, we can add vector 𝐂𝐃. Our expression can be rewritten as 𝐄𝐁 plus 𝐁𝐀 plus 𝐂𝐃.
From the diagram, we can see that the vector 𝐄𝐁 can be rewritten as vector 𝐄𝐀 plus vector 𝐀𝐁. This gives us the expression 𝐄𝐀 plus 𝐀𝐁 plus 𝐁𝐀 plus 𝐂𝐃. As vector 𝐀𝐁 and vector 𝐁𝐀 are the additive inverse of one another, they will sum to give us the zero vector. So, we can therefore cancel these terms. As vector 𝐀𝐄 is equal to vector 𝐄𝐃, then vector 𝐄𝐀 will be equal to vector 𝐃𝐄. Our expression can be simplified to vector 𝐃𝐄 plus vector 𝐂𝐃.
We can change the order of these two vectors so that the endpoint of the first vector is the start point of the second vector. We can see from the diagram that the vector 𝐂𝐃 plus the vector 𝐃𝐄 is equivalent to the vector 𝐂𝐄. The correct answer is therefore option (B).
If 𝐴𝐵𝐶𝐷 is a rectangle and 𝐸 is the midpoint of line segment 𝐴𝐷, then the vector 𝐄𝐁 plus the vector 𝐁𝐀 minus the vector 𝐃𝐂 is equal to the vector 𝐂𝐄.