Video Transcript
π΄π΅πΆπ· is a trapezium. If vector ππ plus vector ππ is equal to π multiplied by vector π±π², then π is equal to blank, where π 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 π± is the midpoint of line segment ππ, then the vector ππ± is equal to the vector π±π. And both of these are equal to a half multiplied by the vector ππ. In the same way, as π² is the midpoint of the line segment ππ, then vector ππ² is equal to vector π²π. And both of these are equal to a half of the vector ππ.
We know that the general vectors ππ and ππ are the additive inverse of each other. This means that the vector ππ plus the vector ππ is equal to the zero vector. This can be rewritten such that the vector ππ is equal to the negative of vector ππ.
Letβs now consider the equation we are given in this question in order to calculate π. We can see from the diagram that the vector ππ is equal to the vector ππ± plus the vector π±π² plus the vector π²π. In the same way, the vector ππ is equal to the vector ππ± plus the vector π±π² plus the vector π²π. We can then replace the vector ππ± with the vector π±π and the vector π²π with the vector ππ². Collecting like terms, we have two π±π² plus π±π plus ππ± plus π²π plus ππ². Vectors π±π and ππ± as well as vectors π²π and ππ² 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 π±π². This is in the form we were looking for. And we can see that π is equal to two. The correct answer is therefore option (D). If π΄π΅πΆπ· is a trapezium and vector ππ plus vector ππ is equal to π multiplied by vector π±π², then π is equal to two.
