# Video: Pack 4 β’ Paper 3 β’ Question 17

Pack 4 β’ Paper 3 β’ Question 17

04:56

### Video Transcript

Consider the diagram. π΄π΅π·, π΄ππΆ, and π΅ππΆ are straight lines, where π΅ is the midpoint of π΄π· and π is the midpoint of π΅πΆ. The vector πΆπ΄ is equal to nine π, the vector πΆπ΅ is equal to six π, and the vector πΆπ is equal to ππ, where π is a constant. πππ· is a straight line. Find the value of π.

Letβs begin by adding the information weβre given about the vectors πΆπ΄, πΆπ΅, and πΆπ onto the diagram. So we have πΆπ΄ is equal to nine π, πΆπ is equal to ππ, and πΆπ΅ is equal to six π. Weβre also told in the question that πππ· is a straight line. This means that the vectors ππ· and ππ must be parallel or phrased another way ππ· is a scalar multiple of ππ.

Letβs see if we can find expressions for these two vectors, starting with ππ. We donβt have a vector for going directly from π to π, but we can go via the point πΆ. The vector ππ is, therefore, equal to ππΆ plus πΆπ.

Now, we havenβt been given the vector ππΆ, but we have been given the vector πΆπ. And as weβre travelling in the opposite direction, this means that we change the sign. So we have that ππ is equal to negative πΆπ plus πΆπ.

Weβre also told in the question that π is the midpoint of π΅πΆ. This means that the vectors πΆπ and ππ΅ are each half of the vector πΆπ΅. As the vector πΆπ΅ is equal to six π, this means that the vectors πΆπ and ππ΅ are each equal to three π. Substituting the vectors for πΆπ and πΆπ into our expression for ππ, we have that ππ is equal to negative ππ plus three π.

Now, letβs consider the vector ππ·. We donβt have a vector that describes how to go directly from π to π·. But we can go via the point π΅. The vector ππ· is equal to ππ΅ plus π΅π·. Now, we already have an expression for the vector ππ΅, itβs three π. So we just need to find the vector π΅π·.

As π΅ is the midpoint of the line π΄π·, the vector π΅π· will be equal to the vector π΄π΅. We can find the vector π΄π΅ by going via the point πΆ. The vector π΄π΅ is equal to π΄πΆ plus πΆπ΅, both of which we have expressions for. The vector πΆπ΅ is six π. And as weβre travelling in the opposite direction from π΄ to πΆ, the vector π΄πΆ is negative nine π. So this gives an expression for π΄π΅. And as π΄π΅ is equal to π΅π·, we also have an expression for π΅π·.

Substituting the vectors for ππ΅ and π΅π· into our expression for ππ·, we have three π minus nine π plus six π. This simplifies to negative nine π plus nine π. And we have our vector for ππ·.

Now, remember we said that ππ· is a scalar multiple of ππ, which means that the vector ππ· is equal to π multiplied by the vector ππ for some number π. Substituting the expressions we found for ππ· and ππ gives negative nine π plus nine π is equal to π multiplied by negative ππ plus three π. We can expand the bracket on the right side of this equation to give negative nine π plus nine π is equal to negative πππ plus three ππ.

Now, remember weβre looking to find the value of π. And to do so, we need to equate the coefficients of π and π on the two sides of this equation. Equating the coefficients of π, we have negative nine is equal to negative ππ. And equating the coefficients of π, we have that nine is equal to three π.

We can solve the second equation to find the value of π by dividing both sides by three. This tells us that π is equal to three.

Finally, we can substitute this value into our first equation, which tells us that negative nine is equal to negative three π. The negatives on each side of the equation will cancel out. And to find the value of π, we need to divide both sides by three. This gives three is equal to π.

The value of π is three.