Three particles P, Q and R are moving along the vectors $\overrightarrow{A}=\widehat{i}+\widehat{j},\overrightarrow{B}=\widehat{j}+\widehat{k}$  and $\overrightarrow{C}=-\widehat{i}+\widehat{j}$  respectively. They strike on point and start to move in different directions. Now particle P is moving normal to the plane which contains vector $\overrightarrow{A}and\overrightarrow{B}.$  Similarly particle Q is moving normal to the plane which contains vector $\overrightarrow{A}and\overrightarrow{C}.$  The angle between the direction of motion of P and Q $co{s}^{-1}\left(\frac{1}{\sqrt[]{x}}\right).$  Then the value of x is_________.

<p>Cross product will represent a vector perpendicular to both the vector.</p><p>Dir. Of Motion of P : <span contenteditable="false"> <math> <mrow> <mo>=</mo> <mrow> <mo> (</mo> <mrow> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo> (</mo> <mrow> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>×</mo> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span><math><mrow><mrow><mrow><mover accent="true"></mover><mover accent="true"></mover></mrow></mrow><mrow><mrow><mover accent="true"></mover><mover accent="true"></mover></mrow></mrow></mrow></math></p><p>= <span contenteditable="false"> <math> <mrow> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> </mrow> </math> </span><math><mrow><mover accent="true"></mover><mover accent="true"></mover></mrow></math></p><p>Dir. Of Motion of Q = <span contenteditable="false"> <math> <mrow> <mrow> <mo> (</mo> <mrow> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo> (</mo> <mrow> <mo>−</mo> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span><math><mrow><mrow><mrow><mover accent="true"></mover><mover accent="true"></mover></mrow></mrow><mrow><mrow><mover accent="true"></mover><mover accent="true"></mover></mrow></mrow></mrow></math></p><p><span contenteditable="false"> <math> <mrow> <mo>=</mo> <mn>2</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mover accent="true"> <mi>a</mi> <mo>→</mo> </mover> <mo>.</mo> <mover accent="true"> <mi>b</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>⇒</mo> <mi>θ</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span></p>

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