In a two dimensional motion, a particle moves from point A (position vector ${\to r}_{1}$ ), to point B (position vector ${\to r}_{2}$ ). If the magnitudes of these vectors are $r_{1} = 3 m$ and $r_{2} = 4 m$ and the angles they make with the x-axis are $θ_{1} = 75^{\circ} and θ_{2} = 15^{\circ}$ , respectively, then the magnitude of the displacement vector (in $m$ ) is

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Solution

Displacement = $- - \to A B,$ angle between $_{1} a n d_{2} : θ = 75^{\circ} - 15^{\circ} = 60^{\circ}$

$A B^{2} = r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} c o s θ$
$= 3^{2} + 4^{2} - 2 \times 3 \times 4 c o s 60^{\circ} = 13$
$\Rightarrow A B = \sqrt{13} m$

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