Assertion -If u and v are unit vectors inclined at an angle - and x is a unit vector bisecting then angle between them- then x - u - v - 2 Reason- If ABC be an isosceles triangle with AB-AC- then vector representing bisector of angle A is given by AD - AB - AC 2-

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Properties of Determinants

Assertion :If...

Question

Assertion :If $\to u$ and $\to v$ are unit vectors inclined at an angle $α$ and $\to x$ is a unit vector bisecting then angle between them, then $\to x = \frac{\to u + \to v}{\sqrt{2}}$ Reason: If $A B C$ be an isosceles triangle with $A B = A C$ , then vector representing bisector of angle $A$ is given by $- - \to A D = \frac{- - \to A B + - - \to A C}{2}$ .

Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

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Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion

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Assertion is true but Reason is false

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Assertion is false but Reason is true

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Solution

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
$R : - - \to A D = \frac{- - \to A B + - - \to A C}{2}$
$\Rightarrow$ Reason is true{Although $∣ ∣ ∣ - - \to A B ∣ ∣ ∣ = ∣ ∣ ∣ - - \to A C ∣ ∣ ∣ = 1 ∣ ∣ ∣ - - \to A B ∣ ∣ ∣ = ∣ ∣ ∣ - - \to A C ∣ ∣ ∣ = 1 ∣ ∣ ∣ - - \to A D ∣ ∣ ∣ \neq 1$ }
$A :$ Since $∣ ∣ \to u ∣ ∣ = ∣ ∣ \to v ∣ ∣ = 1$
$\Rightarrow ∠$ bisector is along $(\to u + \to v)$
$∴$ a unit vector $^x$ along $(\to u + \to v)$ will be $^x = \frac{\to u + \to v}{\sqrt{2}}$ and it follows from $R$

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Similar questions

Q. Assertion :If

u

and

v

are two unit vectors inclined at an angle

α

and

x

is a unit vector bisecting the angle between them then

x = \frac{u + v}{2 cos \frac{α}{2}}

. Reason: If

A B C

be an isosceles triangle with

A B = A C = 1

then vector representing bisector of angle

A

is Given by

A D = \frac{A B + A C}{2}

Q. Assertion :If

^u

and

^v

are unit vectors inclined at an angle

α

and

^a

is a unit vector bisecting the angle between them, then

^a = \frac{^u +^v}{2 cos (α / 2)}

Reason: If

A B C

is an isosceles triangle with

A B = A C = 1

, then the vector representing bisector of angle

A

\frac{\to A B + \to A C}{2}

Q. Assertion: If AD is a median of ∆ABC, then AB + AC > 2AD.
Reason: The angles opposite to equal sides of a triangle are equal.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Q. Assertion :If

\to u

and

\to v

are unit vectors inclined at an angle

α

and

\to x

is a unit vector bisecting the angle between them, then

\to x = \frac{\to u + \to v}{2 sin (α / 2)}

Reason: If

Δ A B C

is an isosceles triangle with

A B = A C = 1,

then the vector representing the bisector of

∠ A

is given by

\to A D = \frac{\to A B + \to A C}{2}

Q. Assertion: Points A(6, 4), B(-4, -6) and C(4, 6) are such that AB =

\sqrt{200}

, BC=

\sqrt{208}

, AC=

\sqrt{8}

, Since AB+BC>AC, points A, B and C from a triangle.

Reason: If

B C^{2} = A B^{2} + A C^{2}

, then

Δ A B C

is a right triangle, right angled at A.

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