Let - b- be unit vectors- If c be a vector such that the angle between - and c is -12- and b-c-2c- then -6 c-2 is equal to

We have, nbsp;b̂=c+2(c×â) ⇒b̂·c=c·c+2(c×â)·c ⇒b̂·c=|c|^2+2[c â c] ⇒b̂·c=|c|^2 ⋯(1) nbsp; And b̂ c=2(c×â) Squaring on both sides, we get |b̂|^2+|c|^2 2b̂ ...

You visited us 3 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard IX

Mathematics

Scalar Triple Product

Let â, b̂ be ...

Question

Let $^a,^b$ be unit vectors. If $\to c$ be a vector such that the angle between $^a$ and $\to c$ is $\frac{π}{12},$ and $^b = \to c + 2 (\to c \times^a)$ , then ${∣ ∣ 6 \to c ∣ ∣}^{2}$ is equal to

6 (3 + \sqrt{3})

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

6 (3 - \sqrt{3})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3 + \sqrt{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

6 (\sqrt{3} + 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $6 (3 + \sqrt{3})$
We have, $^b = \to c + 2 (\to c \times^a)$
$\Rightarrow^b \cdot \to c = \to c \cdot \to c + 2 (\to c \times^a) \cdot \to c$
$\Rightarrow^b \cdot \to c = | \to c |^{2} + 2 [\to c^a \to c]$
$\Rightarrow^b \cdot \to c = | \to c |^{2} \dots (1)$
And $^b - \to c = 2 (\to c \times^a)$
Squaring on both sides, we get
$|^b |^{2} + | \to c |^{2} - 2^b \cdot \to c = 4 | \to c |^{2} |^a |^{2} {sin}^{2} \frac{π}{12}$
$\Rightarrow 1 + | \to c |^{2} - 2 | \to c |^{2} = 4 | \to c |^{2} {(\frac{\sqrt{3} - 1}{2 \sqrt{2}})}^{2}$
$\Rightarrow 1 - | \to c |^{2} = | \to c |^{2} (2 - \sqrt{3})$
$\Rightarrow 1 = | \to c |^{2} (3 - \sqrt{3})$
$\Rightarrow 36 | \to c |^{2} = \frac{36}{3 - \sqrt{3}} = 6 (3 + \sqrt{3})$

Suggest Corrections

Similar questions

Q. Let

^a

and

^b

be two unit vectors such that the angle between them is

\frac{π}{4}

. If

θ

is the angle between the vectors

^a +^b

and

(^a + 2^b + 2 (^a \times^b))

, then the value of

164 {cos}^{2} θ

is equal to :

Q. Let

^a

and

^b

be two unit vectors such that

| (^a +^b) + 2 (^a \times^b) | = 2

. If

θ \in (0, π)

is the angle between

^a

and

^b

, then among the statements:

(S 1) : 2 ∣ ∣^a \times^b ∣ ∣ = ∣ ∣^a - ˆ b ∣ ∣

(S 2) :

The projection of

^a

(^a +^b)

\frac{1}{2}

Q. If

^a

^b

are unit vectors such that

^a +^b

is also a unit vector then the angle between the vectors

^a

and

^b

Q. Let

^a,^b,^c

be unit vectors and

α, β, γ

are angles between the vectors

^a,^b;^b,^c

and

^c,^a

respectively. If

^a +^b +^c

is also a unit vector then

cos α + cos β + cos γ

is equal to

Let $^a,^b a n d^c$ be three unit vectors such that
$^a \times (^b \times^c) = \frac{\sqrt{3}}{2} (^b +^c) .$ If
$^b$ is not parallel to $^c$ , then the angle between $^a$ and $^b$ is

Join BYJU'S Learning Program

Let ^a,^b be unit vectors. If →c be a vector such that the angle between ^a and →c is π12, and ^b=→c+2(→c×^a), then ∣∣6→c∣∣2 is equal to

Let $^a,^b$ be unit vectors. If $\to c$ be a vector such that the angle between $^a$ and $\to c$ is $\frac{π}{12},$ and $^b = \to c + 2 (\to c \times^a)$ , then ${∣ ∣ 6 \to c ∣ ∣}^{2}$ is equal to