Let F be the foot in question. Notice that OF = OA +
k(AB), where k is an unknown fraction.
In other words, OF = a +
k(b−a).
To answer the question, we simply need to find k, so
let's do it.
Since OF is perpendicular to AB, it follows that OF · AB =
0.
So ( a +
k(b−a) ) ·
(b−a) = 0.
Simplify and re-arrange to solve for k. (Your "vector
algebra" is being tested.)
Alternatively, we can make use of vector projection to
solve.
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
Thanks :)