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19 A, B, P, Q and R are five points in a plane. Show that the sum of the vectors AP, AV, AR, PB, QD U U UJI 11. Let O be the centre of a regular hexagon ABCDEF. Find the sum of the vectors OA, OB, OC, OD, OE and OF. dh nrove that:

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Q1
Let O be the centre a regular hexagon ABCDEF Then the magnitude of sum of the vectors ¯¯¯¯¯¯¯¯OA,¯¯¯¯¯¯¯¯OB,¯¯¯¯¯¯¯¯OC,¯¯¯¯¯¯¯¯¯OD,¯¯¯¯¯¯¯¯OE, and ¯¯¯¯¯¯¯¯OF is
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Q2
If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that OA→+OB→+OC→=OD→+OE→+OF→.
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Q3
If O is a point in space, ABC is a triangle and D,E,F are the mid-points of the sides BC,CA and AB respectively of the triangle,prove that $$ \overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC} = \overrightarrow{OD}+\overrightarrow{OE}+\overrightarrow{OF}. $$
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Q4
The points O,A,B,C,D are such that OA=a, OB=b, OC=2a+3b, OD=a+2b.
Given that the length of OA is three times the length of OB. Show that BD and AC are perpendicular.
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Q5
In Figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that OA2+OB2+OC2−OD2−OE2−OF2=AF2+BD2+CE2.
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