### Video Transcript

If π΄π΅πΆπ· is a square and vector ππ plus vector ππ is equal to π multiplied by vector ππ, then π 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 ππ is equal to vector ππ and vector ππ is equal to vector ππ. The points π and π are the midpoints of the line segments π·πΆ and π΅πΆ, respectively. This means that the vector ππ is equal to the vector ππ and both of these are equal to a half of the vector ππ.

In the same way, vector ππ is equal to vector ππ, and these are equal to a half of vector ππ. In this question, we are told that vector ππ plus vector ππ is equal to π multiplied by vector ππ. We will begin by considering the two vectors on the left-hand side. We can see from the diagram that the vector ππ is equal to the vector ππ plus the vector ππ. We can also see that the vector ππ is equal to the vector ππ plus the vector ππ.

We know that the general vector ππ and the vector ππ are the additive inverse of one another. This means that they sum to give the zero vector. This also means that the vector ππ is equal to the negative of vector ππ. We know that vector ππ is equal to vector ππ and also that vector ππ is equal to a half of vector ππ. This means that vector ππ is equal to two multiplied by vector ππ. We also know that the vector ππ is equal to the vector ππ.

Our expression simplifies to two ππ plus ππ plus ππ plus ππ. ππ and ππ are the additive inverse of each other. Therefore, they sum to give a zero vector. We are therefore left with two ππ plus ππ, which gives us three multiplied by the vector ππ. This gives us the expression in the correct form such that π is equal to three. The correct answer is option (C).

If π΄π΅πΆπ· is a square and vector ππ plus vector ππ is equal to π multiplied by vector ππ, then π is equal to three.