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

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Vector

Three particles

P, Q

and

R

are moving along the vectors

A = \hat{i} + \hat{j}, B = \hat{j} + \hat{k}

and

C = - \hat{i} + \hat{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

A

and

B

. Similarly particle

Q

is moving normal to the plane which contains vector

A

and

C

. The angle between the direction of motion of

P

and

Q

cos^{- 1} (\frac{1}{x})

. Then the value of

x

is ____.

Answer: 3

A direction normal to the plane of two vectors is along their cross product, so P moves along

A \times B

and Q along

A \times C

A \times B = (\hat{i} + \hat{j}) \times (\hat{j} + \hat{k}) = \hat{i} - \hat{j} + \hat{k}

A \times C = (\hat{i} + \hat{j}) \times (- \hat{i} + \hat{j}) = 2 \hat{k}

cos θ = \frac{( i ^ - j ^ + k ^ ) \cdot ( 2 k ^ )}{3 \times 2} = \frac{2}{2 3} = \frac{1}{3}

Comparing with

cos^{- 1} (\frac{1}{x})

gives

x = 3

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Integer answer