Question

The values of x for which the angle between \[\vec{a} = 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} , \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}\]  is obtuse and the angle between \[\vec{b}\] and the z-axis is acute and less than \[\frac{\pi}{6}\]  are

Options

• (a) \[x > \frac{1}{2} or x < 0\]

• (b) \[0 < x < \frac{1}{2}\] 

• (c) \[\frac{1}{2} < x < 15\] 

• (d) ϕ 

Answer 1

(b) \[0 < x < \frac{1}{2}\] 

\[\vec{a} = 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} , \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}\] 

Let the angle between vector a and vector b be A. 

\[\therefore \cos A = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right|\left| \vec{b} \right|} = \frac{\left( 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} \right) . \left( 7 \hat{i} - 2 \hat{j} + x \hat{k} \right)}{\left| 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} \right| \left| 7 \hat{i} - 2 \hat{j} + x \hat{k} \right|}\]

\[ = \frac{14 x^2 - 8x + x}{\sqrt{4 x^4 + 16 x^2 + 1}\sqrt{49 + 4 + x^2}}\]

\[ = \frac{14 x^2 - 7x}{\sqrt{4 x^4 + 16 x^2 + 1}\sqrt{53 + x^2}}\]

\[Now, \ ∠  \text{ A is an obtuse angle }. \]

\[ \therefore \cos A < 0\]

\[ \Rightarrow \frac{14 x^2 - 7x}{\sqrt{4 x^4 + 16 x^2 + 1}\sqrt{53 + x^2}} < 0\]

\[ \Rightarrow 14 x^2 - 7x < 0\]

\[ \Rightarrow 2 x^2 - x < 0\]

\[ \Rightarrow x\left( 2x - 1 \right) < 0\]

\[ \Rightarrow x < 0 \text{ and }\  2x - 1 > 0 \text{ or } x > 0\ \text{ and }\ 2x - 1 < 0\]

\[ \Rightarrow x < 0 \text{ and } x > \frac{1}{2} \text{ or } x > 0 \text{ and } x < \frac{1}{2}\]

\[ \Rightarrow x > 0 \text{ and } x < \frac{1}{2} \left( \text{ As there cannot be any number less than zero and greater than } 1/2 \right)\]

\[ \Rightarrow x \in \left( 0, \frac{1}{2} \right) . . . (i)\] 

\[\text{Let the equation of the z} - \text{axis be z} \hat{k} . \]

\[\text{ And let the angle between } \vec{b} \text{ and z } - \text{ axis be B } . \]

\[ \therefore \cos B = \frac{\left( 7 \hat{i} - 2 \hat{j} + x \hat{k} \right) . \left( z \hat{k} \right)}{\left| 7 \hat{i} - 2 \hat{j} + x \hat{k} \right| \left| z \hat{k} \right|}\]

\[ = \frac{xz}{z\sqrt{49 + 4 + x^2}}\]

\[ = \frac{x}{\sqrt{53 + x^2}}\]

\[\text{ Now, angle B is acute and less than } \pi/6 . \]

\[ \therefore 0 < \frac{x}{\sqrt{53 + x^2}} < \cos\frac{\pi}{6}\]

\[ \Rightarrow 0 < x < \frac{\sqrt{3}}{2}\sqrt{53 + x^2} . . . (ii)\]

\[\text{ From } (i) \text{ and } (ii) \text{ we get }\]

\[0 < x < \frac{1}{2}\]