\(A\) is the point \((-1,4)\) and \(B\) is the point \((7,-4)\). (a) Write \(\overrightarrow{AB}\) as a column vector. [1] (b) Find \(|\overrightarrow{AB}|\...

Coordinate work with vectors uses three ideas: the connecting vector is finish minus start, the length comes from Pythagoras, and a point dividing a line in a given ratio is reached by travelling the matching fraction of the connecting vector.

(a) \(\overrightarrow{AB}=\binom{7-(-1)}{-4-4}=\binom{8}{-8}\) [B1]

(b) \(|\overrightarrow{AB}|=\sqrt{8^{2}+(-8)^{2}}=\sqrt{64+64}=\sqrt{128}\) [M1] ft. Take out the largest square factor: \(128=64\times2\), so \(\sqrt{128}=8\sqrt{2}\) [A1] cao, giving \(k=8\).

(c) The midpoint averages the coordinates: \(\left(\frac{-1+7}{2},\frac{4+(-4)}{2}\right)\) [M1] \(=(3,0)\) [A1]

(d) \(AP:PB=3:1\) divides \(AB\) into \(4\) equal parts, with \(P\) three parts along from \(A\):

1. \(\overrightarrow{AP}=\frac{3}{4}\overrightarrow{AB}=\frac{3}{4}\binom{8}{-8}=\binom{6}{-6}\) [M1]
2. Add this to the position of \(A\): \((-1+6,\;4-6)\) [M1] ft
3. \(P\) is \((5,-2)\) [A1]

The fraction is \(\frac{3}{3+1}=\frac{3}{4}\), not \(\frac{3}{1}\); the denominator is the total number of parts. A check is that \(P\) lies beyond the midpoint \((3,0)\) and closer to \(B(7,-4)\), which matches \(AP\) being three times \(PB\).