In the Cartesian plane, O is the origin of the coordinate axes. A person starts at 0 and walks a distance of 3 units in the NORTH-EAST direction and reaches the point P. From P he walks 4 units distance parallel to NORTHWEST direction and reaches the point Q, then the vector OQ in terms i and j $(∡ XOP = 45^{0})$ is

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$OQ = \frac{1}{\sqrt{2}} (- i + 7 j)$

$OQ = \frac{1}{\sqrt{50}} (- i + 7 j)$

$OQ = \frac{1}{\sqrt{10}} (- i + 7 j)$

$OQ = \frac{1}{\sqrt{2}} (i + 7 j)$

answer is A.

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Detailed Solution

$OP = \frac{3}{\sqrt{2}} i + \frac{3}{\sqrt{2}} j \sin 45^{0} = \frac{β - \frac{3}{\sqrt{2}}}{4} \Rightarrow β = \frac{7}{\sqrt{2}} \cos 45^{0} = \frac{\frac{3}{β} - α}{4} \Rightarrow α = \frac{- 1}{\sqrt{2}} ∴ OQ = \frac{- 1}{\sqrt{2}} i + \frac{7}{\sqrt{2}} j$