QJEEAdv18Maa4f6: JEE Advanced 2018 Physics - Class 11 JEEAdv18Maa4f6 Solution: Let a and b be two unit vectors such that a . b = 0. For some…

Question

Verified Solution • 2025–26 Session moderate

Let a and b be two unit vectors such that a . b = 0. For some x, y

\in

R, let

\overrightarrow c = x\overrightarrow a + y\overrightarrow b + \overrightarrow a \times \overrightarrow b

. If |

\overrightarrow c

| = 2 and the vector c is inclined at the same angle

\alpha

to both a and b, then the value of

8{\cos ^2}\alpha

is ..............

Accepted Answer

We have,

\overrightarrow c = x\overrightarrow a + y\overrightarrow b + \overrightarrow a \times \overrightarrow b

and

\overrightarrow a .\overrightarrow b

= 0

|

\overrightarrow a

| = |

\overrightarrow b

| = 1

and |

\overrightarrow c

| = 2

Also, given

\overrightarrow c

is inclined on

\overrightarrow a

and

\overrightarrow b

with same angle

\alpha

.

\therefore

\overrightarrow a .\overrightarrow c = x|\overrightarrow a {|^2} + y(\overrightarrow a .\overrightarrow b ) + \overrightarrow a .(\overrightarrow a \times \overrightarrow b )

|\overrightarrow a ||\overrightarrow c |cos\alpha = x + 0 + 0

x = 2cos

\alpha

Similarly,

|\overrightarrow b ||\overrightarrow c |cos\alpha = 0 + y + 0

\Rightarrow

y = 2cos

\alpha

|\overrightarrow c {|^2} = {x^2} + {y^2} + |\overrightarrow a \, \times \,\overrightarrow b {|^2}

4 = 8{\cos ^2}\alpha + |a{|^2}|b{|^2}{\sin ^2}90^\circ

4 = 8{\cos ^2}\alpha + 1

\Rightarrow

8{\cos ^2}\alpha

= 3

Step-by-Step Verified Solution