𝐴𝐵𝐶𝐷 is a trapezium. If vector 𝐀𝐃 plus vector 𝐁𝐂 is equal to 𝑘 multiplied by vector 𝐱𝐲, then 𝑘 is equal to blank, where 𝑘 is a real number. Is the answer (A) negative two, (B) negative one, (C) one, or (D) two?
We know that two vectors are equal if they have the same direction and magnitude. As 𝐱 is the midpoint of line segment 𝐀𝐁, then the vector 𝐀𝐱 is equal to the vector 𝐱𝐁. And both of these are equal to a half multiplied by the vector 𝐀𝐁. In the same way, as 𝐲 is the midpoint of the line segment 𝐃𝐂, then vector 𝐃𝐲 is equal to vector 𝐲𝐂. And both of these are equal to a half of the vector 𝐃𝐂.
We know that the general vectors 𝐏𝐐 and 𝐐𝐏 are the additive inverse of each other. This means that the vector 𝐏𝐐 plus the vector 𝐐𝐏 is equal to the zero vector. This can be rewritten such that the vector 𝐏𝐐 is equal to the negative of vector 𝐐𝐏.
Let’s now consider the equation we are given in this question in order to calculate 𝑘. We can see from the diagram that the vector 𝐀𝐃 is equal to the vector 𝐀𝐱 plus the vector 𝐱𝐲 plus the vector 𝐲𝐃. In the same way, the vector 𝐁𝐂 is equal to the vector 𝐁𝐱 plus the vector 𝐱𝐲 plus the vector 𝐲𝐂. We can then replace the vector 𝐀𝐱 with the vector 𝐱𝐁 and the vector 𝐲𝐂 with the vector 𝐃𝐲. Collecting like terms, we have two 𝐱𝐲 plus 𝐱𝐁 plus 𝐁𝐱 plus 𝐲𝐃 plus 𝐃𝐲. Vectors 𝐱𝐁 and 𝐁𝐱 as well as vectors 𝐲𝐃 and 𝐃𝐲 are additive inverses. This means they are equal to the zero vector and will therefore cancel.
Our expression simplifies to two multiplied by the vector 𝐱𝐲. This is in the form we were looking for. And we can see that 𝑘 is equal to two. The correct answer is therefore option (D). If 𝐴𝐵𝐶𝐷 is a trapezium and vector 𝐀𝐃 plus vector 𝐁𝐂 is equal to 𝑘 multiplied by vector 𝐱𝐲, then 𝑘 is equal to two.