Question

the magnitude of a vector A is constant but it is changing direction continuously the angle between A and dA/dt


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Answer 1

$\text{It is given that }|\bar{A}| \text{ is constant}$

$\text{Let }|\bar{A}| = A$

$\implies \bar{A}.\bar{A} = A^{2}$

$\text{Differentiating with respect to }t$

$\implies \dfrac{d(\bar{A}\bar{A})}{dt} = \dfrac{d(A^{2})}{dt}$

$\text{We know that }A \text{ is a constant and }\dfrac{d(\text{constant})}{dx} = 0$

$\text{We know that }\dfrac{d(uv)}{dx} = u\text{\huge{(}}\dfrac{dv}{dx}\text{\huge{)}} + v\text{\huge{(}}\dfrac{du}{dx}\text{\huge{)}}$

$\implies \dfrac{d\bar{A}}{dt}.\bar{A} + \dfrac{d\bar{A}}{dt}.\bar{A} = 0$

$\implies 2\dfrac{d\bar{A}}{dt}.\bar{A} = 0$

$\implies \dfrac{d\bar{A}}{dt}.\bar{A} = 0$

$\text{We know that the dot product of two perpendicular vectors is 0}$

$\implies \dfrac{d\bar{A}}{dt} \text{ and }\bar{A} \text{ are perpendicular}$

$\implies \boxed{\text{Angle between }\bar{A} \text{ and }\dfrac{d\bar{A}}{dt} = \dfrac{\pi}{2} = 90^{\circ} }$