ABC is a triangle. M is the midpoint of AB and N is the midpoint of BC. overrightarrow (AM)=3a overrightarrow (AC)=5b a) Express overrightarrow (MN) in terms of a and bin its simplest form. b) Prove that MN is parallel to A C. Hence , calculate how many times longer AC is than MN. Not drawn accurately

<p> Here we are dealing with vectors. The triangle ABC's vertices and directions indicate that it's a Vector Geometry related subject. <br />a) vector AB=AM*2 (as M is a midpoint on AB line); vector AB=vector AC-vector BC. Hence we need to get vectors of AC and BC.<br /> vec (AC)=5b, vec (AM)=3a;<br /> vec (BC)=vec (AB)- vec (AC)= vec (AM)*2- vec (AC)=2*3a-5b=6a-5b.<br />To express vector MN, we remind that M is midpoint of both vector AB and vector AN. So vector MN= vector AN= vector AC/2, we substitute the known values of vec AC; AC=magnitude AC(angle AC), so AC needs to be used with another variable for calculation.<br />b) proving parallelism includes vector division, that produces λ; if λ is inflotation number comparison between magnitude MN and AB while keeping the same directions, it means line MN is parallel to line AB. Following the method, λ=vec MN/ vec AC. <br />The quantity (magnitude AC bundled with direction) is greater compared to vec MN (which includes magnitude and direction). We can express minus "-" before magnitudes (numbers that interpretes length of comparison MN is greater/smaller than AB) only regarding which vector we started at first.