Let and be three non-zero vectors such that and are non-collinear. If is collinear with is collinear with and , then is equal to [2024]

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Solution

Q.
Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a} + 5 \vec{b}$ is collinear with $\vec{c}, \vec{b} + 6 \vec{c}$ is collinear with $\vec{a}$ and $\vec{a} + α \vec{b} + β \vec{c} = \vec{0}$ , then $α + β$ is equal to [2024]

1 35

2 –30

3 30

4 –25

Ans.
(1)

We have

$\vec{a} + 5 \vec{b} = λ \vec{c}, λ \in R$ ... (i)

$\vec{b} + 6 \vec{c} = μ \vec{a}, μ \in R \Rightarrow \vec{a} = \frac{\vec{b} + 6 \vec{c}}{μ}$

From (i), $\frac{\vec{b} + 6 \vec{c}}{μ} + 5 \vec{b} = λ \vec{c} \Rightarrow \vec{b} (\frac{1}{μ} + 5) = \vec{c} (λ - \frac{6}{μ})$

Since, $\vec{b}$ and $\vec{c}$ are non-collinear.

$∴ \frac{1}{μ} = - 5 and λ = \frac{6}{μ} \Rightarrow μ = \frac{- 1}{5} and λ = - 30$

$∴ From (i), \vec{a} + 5 \vec{b} - λ \vec{c} = 0$

$\Rightarrow \vec{a} + 5 \vec{b} + 30 \vec{c} = 0 \Rightarrow α = 5 and β = 30 ∴ α + β = 35$

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