A woman runs 12 km towards her North, then 6 km towards her South and then 8 km towards her East. In which direction is she from her starting point?

Correct Option (2)

The woman's final position is determined by vector addition of her movements:

• Initial movement: 12 km North. Position: (0, 12).
• Second movement: 6 km South. Net Northward displacement: 12 - 6 = 6 km. Position: (0, 6).
• Third movement: 8 km East. Final position: (8, 6).

The net displacement from the starting point is 8 km East and 6 km North. This places her in a direction North of East.

The angle (θ) with respect to the East axis is given by:

tan(θ) = (Vertical displacement) / (Horizontal displacement) = 6/8 = 0.75.

Given tan(45°) = 1, and tan(θ) = 0.75, it follows that θ < 45°.

Thus, her position is at an angle less than 45° North of East from the starting point.

Incorrect Options:

• An angle less than 45° South of East: Incorrect, as the net vertical displacement is 6 km North, not South.
• An angle more than 45° South of East: Incorrect. The displacement is North, not South, and the calculated angle is less than 45°.
• An angle more than 45° North of East: Incorrect. While the direction is North of East, the calculated angle (θ) is less than 45° because tan(θ) = 0.75 < tan(45°).