Video Transcript
If 𝐴𝐵𝐶𝐷 is a square and vector 𝐀𝐘 plus vector 𝐗𝐘 is equal to 𝑘 multiplied by vector 𝐗𝐂, then 𝑘 is equal to blank. Is it (A) one, (B) two, (C) three, or (D) four?
We recall that two vectors are equal if they have the same direction and magnitude. This means that when dealing with a square, the parallel sides will be represented by the same vectors as they have the same magnitude and direction. In this question, this means that vector 𝐀𝐁 is equal to vector 𝐃𝐂 and vector 𝐀𝐃 is equal to vector 𝐁𝐂. The points 𝑋 and 𝑌 are the midpoints of the line segments 𝐷𝐶 and 𝐵𝐶, respectively. This means that the vector 𝐃𝐗 is equal to the vector 𝐗𝐂 and both of these are equal to a half of the vector 𝐃𝐂.
In the same way, vector 𝐁𝐘 is equal to vector 𝐘𝐂, and these are equal to a half of vector 𝐁𝐂. In this question, we are told that vector 𝐀𝐘 plus vector 𝐗𝐘 is equal to 𝑘 multiplied by vector 𝐗𝐂. We will begin by considering the two vectors on the left-hand side. We can see from the diagram that the vector 𝐀𝐘 is equal to the vector 𝐀𝐁 plus the vector 𝐁𝐘. We can also see that the vector 𝐗𝐘 is equal to the vector 𝐗𝐂 plus the vector 𝐂𝐘.
We know that the general vector 𝐏𝐐 and the vector 𝐐𝐏 are the additive inverse of one another. This means that they sum to give the zero vector. This also means that the vector 𝐏𝐐 is equal to the negative of vector 𝐐𝐏. We know that vector 𝐀𝐁 is equal to vector 𝐃𝐂 and also that vector 𝐗𝐂 is equal to a half of vector 𝐃𝐂. This means that vector 𝐀𝐁 is equal to two multiplied by vector 𝐗𝐂. We also know that the vector 𝐁𝐘 is equal to the vector 𝐘𝐂.
Our expression simplifies to two 𝐗𝐂 plus 𝐘𝐂 plus 𝐗𝐂 plus 𝐂𝐘. 𝐘𝐂 and 𝐂𝐘 are the additive inverse of each other. Therefore, they sum to give a zero vector. We are therefore left with two 𝐗𝐂 plus 𝐗𝐂, which gives us three multiplied by the vector 𝐗𝐂. This gives us the expression in the correct form such that 𝑘 is equal to three. The correct answer is option (C).
If 𝐴𝐵𝐶𝐷 is a square and vector 𝐀𝐘 plus vector 𝐗𝐘 is equal to 𝑘 multiplied by vector 𝐗𝐂, then 𝑘 is equal to three.
