Question

Let `hata` and `hatb` be two unit vectors such that the angle between them is `π/4`. If θ is the angle between the vectors `(hata + hatb)` and `(hata xx 2hatb + 2(hata xx hatb))`, then the value of 164 cos²θ is equal to ______.

पर्याय

• `90 + 27sqrt(2)`

• `45 + 18sqrt(2)`

• `90 + 3sqrt(2)`

• `54 + 90sqrt(2)`

Answer 1

Let `hata` and `hatb` be two unit vectors such that the angle between them is `π/4`. If θ is the angle between the vectors `(hata + hatb)` and `(hata xx 2hatb + 2(hata xx hatb))`, then the value of 164 cos²θ is equal to `underlinebb(90 + 27sqrt(2))`.

Explanation:

Here we are given that angle between `hata` and `hatb`

= `π/4` = `phi`

`hata.hatb = |hata||hatb|cos phi`

Now, `hata.hatb = cos phi = 1/sqrt(2)`

Now, cos θ = `((hata + hatb).(hata + 2hatb + 2(hata xx hatb)))/(|hata + hatb||hata + 2hatb + 2(hata xx hatb)|`

`\implies |hata + hatb|^2 = (hata + hatb).(hata + hatb)`

`\implies |hata + hatb|^2 = 2 + 2hata.hatb = 2 + sqrt(2)`

Now `hata xx hatb = |hata||hatb| sinphi hatn`

`\implies hata xx hatb = hatn/sqrt(2)` when `hatn` is vector ⊥ `hata` and `hatb`

Let `vecc = hata xx hatb`

We know `vecc.vecb` = 0; `vecc.vecb` = 0

`\implies |hata + 2hatb + 2vecc|^2`

= `1 + 4 + ((4))/2 + 4hata.hatb + 8hatb.vecc + 4vecc.hata`

= `7 + 4/sqrt(2)`

= `7 + 2sqrt(2)`

Now

`(hata + hatb).(hata + 2hatb + 2vecc)`

= `|hata|^2 + 2hata.hatb + 0 + hatb.hata + 2|hatb|^2 + 0`

= `1 + 2/sqrt(2) + 1/sqrt(2) + 2`

= `3 + 3/sqrt(2)`

Now cos θ = `(3 + 3/sqrt(2))/(sqrt(2 + sqrt(2))sqrt(7 + 2sqrt(2))`

`\implies` cos² θ = `(9(sqrt(2) + 1)^2)/(2(2 + sqrt(2))(7 + 2sqrt(2))`

`\implies` cos² θ = `(9/(2sqrt(2)))((sqrt(2) + 1))/((7 + 2sqrt(2)))`

`\implies` 164 cos² θ = `((82)(9)(sqrt(2) + 1)(7 - 2sqrt(2)))/((7 + 2sqrt(2))(7 - 2sqrt(2))`

= `((82))/sqrt(2) ((9)[7sqrt(2) - 4 + 7 - 2sqrt(2)])/((41))`

= `(9sqrt(2))[5sqrt(2) + 3]`

= `90 + 27sqrt(2)`