Question
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
Answer
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Gregory
Professional · Tutor for 6 years
Answer
a) vec AB=AM*2; vec BC= vec AM*2- vec AC=6a-5b; vec MN= vec AN= vec AC/2=5b/2=2.5b.b) Mn || AB only if vec MN/ vec AB if and only if it is real positive number-so parallel. λ=vec MN/ vec AC. λ=2.5/5=0.5 Therefore, MN is parallel to AB. As specified by the λ results-of magnitude, the number that vec AC greater as vev MN: AC greater than 2 times than MN.AppBundle\Command\Annotation\× PhpName M Invoice $client + Look-up Trae'
Explanation
Here we are dealing with vectors. The triangle ABC's vertices and directions indicate that it's a Vector Geometry related subject. 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. vec (AC)=5b, vec (AM)=3a; vec (BC)=vec (AB)- vec (AC)= vec (AM)*2- vec (AC)=2*3a-5b=6a-5b.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.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. 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.