If the vectors a = i ^ + j ^ + 4 k ^ , b = - 2 i ^ + 4 j ^ - 2 k ^ and c = 2 i ^ + 3 j ^ + k ^ are coplanar and the projection of a on the vector b is 54 units, then the sum of all possible values of is equal to [2023]

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Solution

Q.
If the vectors $\vec{a} = λ \hat{i} + μ \hat{j} + 4 \hat{k}, \vec{b} = - 2 \hat{i} + 4 \hat{j} - 2 \hat{k}$ and $\vec{c} = 2 \hat{i} + 3 \hat{j} + \hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $λ + μ$ is equal to [2023]

1 18

2 6

3 24

4 0

Ans.
(3)

$Given \vec{a}, \vec{b}, \vec{c} are coplanar, so$

$| \begin{matrix} λ & μ & 4 \\ - 2 & 4 & - 2 \\ 2 & 3 & 1 \end{matrix} |$ $= 0 \Rightarrow 10 λ - 2 μ = 56$ ...(i)

$Also, projection of \vec{a} on the vector \vec{b} is \sqrt{54}, so,$

$\frac{\vec{a} \cdot \vec{b}}{| \vec{b} |} = \frac{- 2 λ + 4 μ - 8}{\sqrt{24}} = \sqrt{54} \Rightarrow - λ + 2 μ = 22$ ...(ii)

or $- λ + 2 μ = - 14$ ...(iii)

$Solving equations (i) and (ii), we get$

$λ = \frac{26}{3} and μ = \frac{46}{3}$

$Solving equation (i) and (iii), we get, λ = \frac{14}{3}$ and $μ = \frac{- 14}{3}$

$So, sum of all possible values of λ + μ = \frac{26}{3} + \frac{46}{3} = 24$

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