First slide

Direction ratios and direction cosines

Question

If origin is the orthocentre of a triangle formed by the options $(\cos α, \sin α, 0), (\cos β, \sin β, 0), (\cos γ, \sin γ, 0)$ then $\sum \cos (2 α - β - γ) = -$

Moderate

A

0
B

1
C

2
D

3

Solution

$O A = O B = O C; G = H = O (0, 0, 0)$
Equilateral triangle
$\cos α + c o s β = - c o s γ, \sin α + \sin β = - \sin γ,$
square and add $\cos (α - β) = - \frac{1}{2}$
$\cos (β - γ) = \cos (γ - α)$
$\cos (2 α - β - γ) = \cos [(α - β) - (γ - α)] = 1$

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