The set of all , for which the vectors and are inclined at an obtuse angle for all , is [2024]

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Solution

Q.
The set of all $α$ , for which the vectors $\vec{a} = α t \hat{i} + 6 \hat{j} - 3 \hat{k}$ and $\vec{b} = t \hat{i} - 2 \hat{j} - 2 α t \hat{k}$ are inclined at an obtuse angle for all $t \in R$ , is [2024]

1 [0, 1)

2 (–2, 0]

3 $(- \frac{4}{3}, 0]$

4 $(- \frac{4}{3}, 1)$

Ans.
(3)

Given, $\vec{a} = α t \hat{i} + 6 \hat{j} - 3 \hat{k}$ and $\vec{b} = t \hat{i} - 2 \hat{j} - 2 α t \hat{k}$

So, $\vec{a} \cdot \vec{b} < 0 \forall t \in R$ [ $∴$ $\vec{a}$ and $\vec{b}$ are inclined at an obtuse angle]

$\Rightarrow α t^{2} - 12 + 6 α t < 0 \forall t \in R \Rightarrow α < 0 and D < 0$

$\Rightarrow 36 α^{2} + 48 α < 0 \Rightarrow 12 α (3 α + 4) < 0$

$\Rightarrow \frac{- 4}{3} < α < 0$

Also, for $α = 0, \vec{a} \cdot \vec{b} < 0 \Rightarrow α \in (- \frac{4}{3}, 0]$ .

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