1You walk from your home A to a shop B and then from B to your school C. What is the total distance, and is it equal to, less than or greater than the straight distance AC?
2Thinking of distance and direction together: you went from A to B and then from B to C. Where did you start and where did you finish? Why is \( \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \)?
3Prove that in triangle ABC, if D is the mid-point of BC, then \( \overrightarrow{AD} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC}) \).
4Find the position vector of the mid-point of the line segment joining A(4, -2) and B(-6, 8).
5Prove that the position vector of the point P dividing AB internally in the ratio m : n is \( \vec{p} = \frac{n\vec{a} + m\vec{b}}{m+n} \).
6Find the point that divides the line segment joining A(1, 2) and B(7, 8) internally in the ratio 1 : 2.
7Find the point that divides the line segment joining A(1, 1) and B(4, 4) externally in the ratio 2 : 1.
8Show that the internal division formula gives the mid-point formula when the ratio is 1 : 1. Why is there no external division in the ratio 1 : 1?
9In triangle ABC, E is the mid-point of CA. Write the median vector \( \overrightarrow{BE} \) in terms of the side vectors, and explain why both side vectors start at B.