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A lvl H2 Maths: Vectors

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Relative to an origin O, the position vectors of points A and B are a and b respectively. Givent that angle AOB is 90deg, show that the position vector of the foot of the perpendicular from O to AB is

a + (|a|2 / (|a|2 + |b|2)) (b - a)


Using the concept of vector projection, (i.e. AF = |AF| * unit vector AB),

OF = OA + AF = a + |a| cos θ * (b - a) / |b - a|, where θ = angle between OA and BA
= a + |a| [ {a.(a - b)} / {|a| |a - b| }] * (b - a) / |b - a|
= a + {a.a - a.b} / (|a|2 + |b|2) * (b - a)
= a + |a|2 / (|a|2 + |b|2) * (b - a) since a.a = |a|2 and a.b = 0


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