If -> a = 2hat i + hat j + 3hat k, -> b = 3hat i + 3hat j + hat k and -> c = c_1hat i + c_2hat j + c_3hat k are coplanar vectors and -> a . -> c = 5, -> b ⊥ -> c , then 122(c_1 + c_2 + c_3) is equal to ___________.

2C_1 + C_2 + 3C_3 = 5 ...... (i) 3C_1 + 3C_2 + C_3 = 0 ...... (ii) [vec a vec b vec c] = 2 1 3 \ 3 3 1 \ C_1 C_2 C_3 \ = 2(3C_3 - C_2) - 1(3C_3 - C_1) + 3(3C_2 - 3C_1) = 3C_3 + 7C_2 - 8C_1 => 8C_1 - 7C_2 - 3C_3 = 0 ...... (iii) C_1 = 10 / 122,C_2 = - 85 / 122,C_3 = 225 / 122 So 122(C_1 + C_2 + C_3) = 150

If a 2 i j 3 k b 3 i 3 j k and c c1 i c2 j c3 k are cop... | null (Mathematics)

2{C_1} + {C_2} + 3{C_3} = 5 \quad ...... (i)

3{C_1} + 3{C_2} + {C_3} = 0 \quad ...... (ii)

[\vec{a} \ \vec{b} \ \vec{c}] = \begin{vmatrix} 2 & 1 & 3 \\ 3 & 3 & 1 \\ C_1 & C_2 & C_3 \end{vmatrix}

= 2(3{C_3} - {C_2}) - 1(3{C_3} - {C_1}) + 3(3{C_2} - 3{C_1})

= 3{C_3} + 7{C_2} - 8{C_1}

\Rightarrow 8{C_1} - 7{C_2} - 3{C_3} = 0 \quad ...... (iii)

{C_1} = {{10} \over {122}},{C_2} = {{ - 85} \over {122}},{C_3} = {{225} \over {122}}

\text{ So } 122({C_1} + {C_2} + {C_3}) = 150

Explanation

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