First slide

Dot product or scalar product of vectors

Question

$If \vec{a} = x \vec{i} + (x - 1) \vec{j} + \vec{k} and \vec{b} = (x + 1) \vec{i} + \vec{j} + a \vec{k} always make an acute angle for all x \in R,then the least integral value of a is$

Moderate

A

3
B

4
C

5
D

6

Solution

$\begin{array}{l} \vec{a} . \vec{b} = [x \vec{i} + (x - 1) \vec{j} + \vec{k}] \cdot [(x + 1) \vec{i} + \vec{j} + a \vec{k}] = x (x + 1) + x - 1 + a = x^{2} + 2 x + a - 1 > 0, \forall x \in R, \\ \Rightarrow a > 2 \end{array}$

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