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(B) Let the origin be $O$. The vector $b$ is in the north-east direction,making an angle of $45^{\circ}$ with the east axis. The vector $c$ is in the

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$4\sqrt{2}$,towards west

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(B) Let the origin be $O$. The vector $b$ is in the north-east direction,making an angle of $45^{\circ}$ with the east axis. The vector $c$ is in the north-west direction,making an angle of $135^{\circ}$ with the east axis.Since the angle between $b$ and $c$ is $90^{\circ}$,the vectors are perpendicular,i.e.,$b \cdot c = 0$.Given $|b| = 4$ and $|c| = 4$.We need to find the magnitude of $d = c - b$:$|d|^2 = |c - b|^2 = |c|^2 + |b|^2 - 2(c \cdot b) = 4^2 + 4^2 - 0 = 16 + 16 = 32$.Thus,$|d| = \sqrt{32} = 4\sqrt{2}$.To find the direction,consider the components:$b = 4(\cos 45^{\circ} \hat{i} + \sin 45^{\circ} \hat{j}) = 4(\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}) = 2\sqrt{2} \hat{i} + 2\sqrt{2} \hat{j}$.$c = 4(\cos 135^{\circ} \hat{i} + \sin 135^{\circ} \hat{j}) = 4(-\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}) = -2\sqrt{2} \hat{i} + 2\sqrt{2} \hat{j}$.Then $d = c - b = (-2\sqrt{2} - 2\sqrt{2}) \hat{i} + (2\sqrt{2} - 2\sqrt{2}) \hat{j} = -4\sqrt{2} \hat{i} + 0 \hat{j}$.Since the $x$-component is negative and the $y$-component is zero,the vector $d$ points towards the west.

The vectors $b$ and $c$ are in the direction of north-east and north-west respectively and $|b|=|c|= 4$. The magnitude and direction of the vector $d = c - b$ are:

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