First slide

Vectors

Question

If vectors $\vec{A} = cosωt \hat{i} + sinωt \hat{j} and \vec{B} = \cos \frac{ωt}{2} \hat{i} + \sin \frac{ωt}{2} \hat{j}$ are functions of time, then the value of t at which they are orthogonal to each other is:

Moderate

A

t = 0
B

$t = \frac{π}{4 ω}$
C

$t = \frac{π}{2 ω}$
D

$t = \frac{π}{ω}$

Solution

$\vec{A} = cosωt \hat{i} + sinωt \hat{j}$
$\vec{B} = \cos (\frac{ωt}{2}) \hat{i} + \sin (\frac{ωt}{2}) \hat{j}$
For $\vec{A} and \vec{B} orthogonal \vec{A .} \vec{B} = 0$
$(cosωt \hat{i} + sinωt \hat{j}) . (\cos \frac{ωt}{2} \hat{i} + \sin \frac{ωt}{2} \hat{j}) = 0$
$cosωt . \cos \frac{ωt}{2} + sinωt . \sin \frac{ωt}{2} = 0$
$\cos (ωt - \frac{ωt}{2}) = 0$
$\Rightarrow \cos \frac{ωt}{2} = 0$
$\frac{ωt}{2} = \frac{π}{2} \Rightarrow ωt = π \Rightarrow t = \frac{π}{ω}$

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