First slide

Dot product or scalar product of vectors

Question

$Let \vec{A} and \vec{B} be two non-parallel unit vectors in a plane. If (α \vec{A} + \vec{B}) bisects the$ $internal angle between \vec{A} and \vec{B}, then \propto is equal to$

NA

A

$\frac{1}{2}$
B

$1$
C

$2$
D

$4$

Solution

$\vec{A} \cdot (α \vec{A} + \vec{B}) = \vec{B} \cdot (α \vec{A} + \vec{B}) \Rightarrow a s |\vec{A}| = |\vec{B}|$
$α + \vec{A} \cdot \vec{B} = α \vec{A} \cdot \vec{B} + 1$
$(α - 1) (\vec{A} \cdot \vec{B} - 1) = 0$
$\Rightarrow \vec{A} \cdot \vec{B} \neq 1 as \vec{A} \cdot \vec{B} = |\vec{A}| \cdot |\vec{B}| cos θ then θ=0 which means parallel but given non parallel \Rightarrow \propto = 1$

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