Vector changes Geometry to Algebra
1. No complexity of Analytical Geometry
2. Remove the astute dotted (helping) line in Geometry
3. No need diagram: Use only 2 vector properties:
Head- to-Tail:
$latex vec{AC}=vec{AB}+vec {BC}$
Closed Loop:
$latex vec{DE}+vec{EF}+vec{FD}=0$
4. Enable Computer automated proof of Geometry via Algebra.
Example: 任意四边形 Quadrilateral ABCD with M,N midpoints of AB, CD, resp.
Prove： MN=1/2(BC+AD)
Proof: (by vector):
Consider MBCN:
MN=MB+ BC+ CN..(1)
Consider MADN:
MN=MA+ AD+ DN..(2)
(1) +(2):
2MN=(MB +MA) +
(BC +AD) +(CN +DN)
but (MB +MA) =0,
(CN +DN) =0 [same magnitude but different direction cancelled out ]
=> MN=1/2 (BC +AD)
Special cases:
1. A = B (=M)
=> triangle ACD
AN = 1/2 (AC +AD)
2. BC // AD
=> Trapezium ABCD
MN=1/2 (BC +AD)
=> MN // BC // AD
Mathtuition88 