An ant starts from the origin and crawls 10 | vedclass

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(A) The ant's displacement vector $\vec{a}$ is given by the movement along the $x$ and $y$ axes:$\vec{a} = 10 \hat{i} + 20 \hat{j}$The position vector

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$30 \ cm^2$

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(A) The ant's displacement vector $\vec{a}$ is given by the movement along the $x$ and $y$ axes:$\vec{a} = 10 \hat{i} + 20 \hat{j}$The position vector $\vec{b}$ of the point with magnitude $r = \sqrt{2} \ cm$ at an angle $	heta = 45^{\circ}$ with the $x$-axis is:$\vec{b} = r \cos 	heta \hat{i} + r \sin 	heta \hat{j}$$\vec{b} = \sqrt{2} \cos 45^{\circ} \hat{i} + \sqrt{2} \sin 45^{\circ} \hat{j}$$\vec{b} = \sqrt{2} \left( \frac{1}{\sqrt{2}} ight) \hat{i} + \sqrt{2} \left( \frac{1}{\sqrt{2}} ight) \hat{j} = \hat{i} + \hat{j}$The dot product $\vec{a} \cdot \vec{b}$ is:$\vec{a} \cdot \vec{b} = (10 \hat{i} + 20 \hat{j}) \cdot (\hat{i} + \hat{j})$$= (10 	imes 1) + (20 	imes 1) = 10 + 20 = 30 \ cm^2$

An ant starts from the origin and crawls $10 \ cm$ along the $x$-axis and then $20 \ cm$ along the $y$-axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45^{\circ}$ with the $x$-axis and has a magnitude of $\sqrt{2} \ cm$ is

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