Let $\overrightarrow{a}and\overrightarrow{b}$ be the vectors along the diagonals of a parallelogram having area $2\sqrt[]{2}.$ Let the angle between $\overrightarrow{a}and\overrightarrow{b}$ be acute, $\left|\overrightarrow{a}\right|=1,and\left|\overrightarrow{a}.\overrightarrow{b}\right|=\left|\overrightarrow{a}\times \overrightarrow{b}\right|.$ If $\overrightarrow{c}=2\sqrt[]{2}\left(\overrightarrow{a}\times \overrightarrow{b}\right)-2\overrightarrow{b},$ then an angle between $\overrightarrow{b}and\overrightarrow{c}is$

<p>Be the vectors along the diagonals of a parallelogram having are 2.</p><p><math><mrow><mo>∴</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>|</mo><mrow><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>b</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow><mo>=</mo><mn>2</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot></mrow></math></p><p><math><mrow><mrow><mo>|</mo><mrow><mover accent="true"><mi>a</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow><mrow><mo>|</mo><mrow><mover accent="true"><mi>b</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow><mi>s</mi><mi>i</mi><mi>n</mi><mi>θ</mi><mo>=</mo><mn>4</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot></mrow></math></p><p><math><mrow><mo>⇒</mo><mrow><mo>|</mo><mrow><mover accent="true"><mi>b</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow><mi>s</mi><mi>i</mi><mi>n</mi><mi>θ</mi><mo>=</mo><mn>4</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot></mrow></math>&nbsp;……. (i)</p><p>And</p><p><math><mrow><mo>⇒</mo><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>.</mo><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>=</mo><mo>−</mo><mn>2</mn><msup><mrow><mrow><mo>|</mo><mrow><mover accent="true"><mi>b</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mo>−</mo><mn>1</mn><mn>2</mn><mn>8</mn><mtext></mtext><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false"> (</mo><mi>i</mi><mi>i</mi><mo stretchy="false">)</mo></mrow></math></p><p><math><mrow><mo>⇒</mo><mrow><mo>|</mo><mrow><mover accent="true"><mi>c</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow><mo>=</mo><mn>1</mn><mn>6</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false"> (</mo><mi>i</mi><mi>i</mi><mi>i</mi><mo stretchy="false">)</mo></mrow></math></p><p>From (ii) and (iii)</p><p><math><mrow><mrow><mo>|</mo><mrow><mover accent="true"><mi>c</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow><mrow><mo>|</mo><mrow><mover accent="true"><mi>b</mi><mo>→</mo></mover></mrow><mo>|</mo></mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>α</mi><mo>=</mo><mo>−</mo><mn>1</mn><mn>2</mn><mn>8</mn></mrow></math></p><p><math><mrow><mo>⇒</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>α</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot></mrow></mfrac></mrow></math></p><p><math><mrow><mi>α</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></p>

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