Given the problem involves finding the value of
(α+β+γ)2 when
(α,β,γ) are the direction cosines of the angular bisector of two lines with direction ratios
(2,2,1) and
(2,−1,−2).
Finding Direction Cosines:
The direction ratios for line
L1 are
(2,2,1), and for line
L2 are
(2,−1,−2).
The direction cosines for
L1 are found as:
(‌,‌,‌)Similarly, for
L2 the direction cosines are:
(‌,‌,‌)Perpendicular Check:
The dot product of the two direction ratios confirms that the lines are perpendicular:
2×2+2×(−1)+1×(−2)=0Angle Bisector Direction Cosines:
The formulas for the direction cosines of the angle bisectors are:
(‌,‌,‌)and
(‌,‌,‌)Calculate the Direction Cosines:
For an angle bisector direction, calculate:
(‌,‌,‌)Another possible set is:
(0,‌,‌)Calculation of
(α+β+γ)2 :
Using
α=‌,β=‌,γ=‌ :
Sum:
α+β+γ=‌+‌−‌=‌ (α+β+γ)2=(‌)2=‌=‌Alternatively, use
α=0,β=‌,γ=‌ :
Sum:
α+β+γ=0+‌+‌=√2Squaring:
(α+β+γ)2=(√2)2=2Based on calculations with valid assumptions, the value of
(α+β+γ)2 considering a correct scenario for calculation comes out to be 2 .