If two sides of a triangle are represented by the vectors -3-b and b-b- where b is a non-zero vector and - is a unit vector in the direction of a- then the angles of triangle are

Let V_=√(3)(â×b) and V_2=b (â·b)â=(â×b)×â V_1·V_2=√(3)(â×b)·[(â×b)×â] ⇒V_1·V_2=0 So, the angle between V_1 and V_2 is ∠ A=90^∘ Now, using sine law in Δ ABC, | ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Physics

Introduction

If two sides ...

Question

If two sides of a triangle are represented by the vectors $\sqrt{3} (^a \times \to b)$ and $\to b - (^a \cdot \to b)^a$ where $\to b$ is a non-zero vector and $^a$ is a unit vector in the direction of $\to a,$ then the angles of triangle are

{tan}^{- 1} (\frac{1}{\sqrt{3}}); {tan}^{- 1} (\frac{1}{2}); {tan}^{- 1} (\frac{\sqrt{3} + 2}{1 - 2 \sqrt{3}})

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

{tan}^{- 1} (\sqrt{3}); {tan}^{- 1} (\frac{1}{\sqrt{3}}); {cot}^{- 1} (0)

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

{tan}^{- 1} (\sqrt{3}); {tan}^{- 1} (2); {tan}^{- 1} (\frac{\sqrt{3} + 2}{2 \sqrt{3} - 1})

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

{tan}^{- 1} (1); {tan}^{- 1} (1); {cot}^{- 1} (0)

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

Open in App

Solution

The correct option is B ${tan}^{- 1} (\sqrt{3}); {tan}^{- 1} (\frac{1}{\sqrt{3}}); {cot}^{- 1} (0)$
Let $\to V = \sqrt{3} (^a \times \to b)$
and $_{2} = \to b - (^a \cdot \to b)^a = (^a \times \to b) \times^a$
$_{1} \cdot_{2} = \sqrt{3} (^a \times \to b) \cdot [(^a \times \to b) \times^a]$
$\Rightarrow_{1} \cdot_{2} = 0$
So, the angle between $_{1}$ and $_{2}$ is $∠ A = 90^{\circ}$

Now, using sine law in $Δ A B C,$
$\frac{∣ ∣ ∣ \to b - (^a \cdot \to b)^a ∣ ∣ ∣}{sin θ} = \frac{\sqrt{3} ∣ ∣ ∣^a \times \to b ∣ ∣ ∣}{cos θ}$
$\Rightarrow tan θ = \frac{1}{\sqrt{3}} \frac{| \to b - (^a \cdot \to b)^a |}{| \to a \times \to b |}$
$\Rightarrow tan θ = \frac{1}{\sqrt{3}} \frac{| (^a \times \to b) \times^a |}{| \to a \times \to b |}$
$\Rightarrow tan θ = \frac{1}{\sqrt{3}} \frac{|^a \times \to b | | \to a | sin 90^{\circ}}{| \to a \times \to b |}$
$\Rightarrow tan θ = \frac{1}{\sqrt{3}}$
$∴ θ = \frac{π}{6}$
The angles of $Δ A B C$ are $\frac{π}{3}, \frac{π}{6}, \frac{π}{2}$ or ${tan}^{- 1} (\sqrt{3}); {tan}^{- 1} (\frac{1}{\sqrt{3}}); {cot}^{- 1} (0)$

Suggest Corrections

Similar questions

Q. The angles of a triangle two of whose sides are represented by the vectors

\sqrt{3} (¯ a \times ¯ b)

and

¯ b - (^a . ¯ b)^a

where

¯ b

is a non-zero vector and

^a

is a unit vector in the direction of

¯ a

are

Q. Let

\to a

be a unit vector and

\to b

be a non-zero vector not parallel to

\to a

. If two sides of a triangle are represented by the vectors

\sqrt{3} (\to a \times \to b)

and

\to b - (\to a \cdot \to b) \to a

, then the angles of the triangle are

Q. The angles of a triangle, two of whose sides are represented by the vectors

\sqrt{3} (\to a \times \to b)

and

\to b - (\to a . \to b) \to a

where

\to b

is a non-zero vector and

\to a

is a unit vector are

Q. If

\to p, \to q

are two non collinear and non zero vectors such that

(b - c) \to p \times \to q + (c - a) \to p + (a - b) \to q = 0

where a, b, c are the lengths of the sides of a triangle, then the triangle is

Q. Let

\to a

and

\to b

are two non-zero, non-collinear vectors

(| \to a | = 1)

such that vectors

3 (\to a \times \to b)

and

2 (\to b - (\to a . \to b) \to a)

represents two sides of a triangle. If area of triangle is

\frac{3}{4} (| \to b |^{4} + 4)

then the value of

| \to b |^{2}

Join BYJU'S Learning Program

If two sides of a triangle are represented by the vectors √3(^a×→b) and →b−(^a⋅→b)^a where →b is a non-zero vector and ^a is a unit vector in the direction of →a, then the angles of triangle are

If two sides of a triangle are represented by the vectors $\sqrt{3} (^a \times \to b)$ and $\to b - (^a \cdot \to b)^a$ where $\to b$ is a non-zero vector and $^a$ is a unit vector in the direction of $\to a,$ then the angles of triangle are