A farmer wants to wall off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in the figure, where A = 4.99 km, B = 2.46 km, and C = 3.23 km and then correctly calculates the length and orientation of D?

Polar Representation: `vec V = (V, \theta)`;
All angles must be measured counter-clockwise from the positive direction of the X-axis (usually East)

`vec A = (4.99` `km, 360^o-7.5^o) = (4.99` `km, 352.5^o);`
`vec B = (2.46` `km, 90^o + 16^o) = (2.46` `km, 106^o);`
`vec C = (3.23` `km, 180^o - 19^o = (3.23` `km, 161^o);`

Cartesian Representation: `vec V = (V_x, V_y) = (V\cos\theta, V\sin\theta);`
`vec A = (4.99\times\cos352.5^o` `km, 4.99\times \sin352.5^o` `km);`
`\qquad = (4.95` `km, -0.65` `km);`
`vec B = (2.46\cos106^o` `km, 2.46\sin106^o` `km);`
`\qquad = (-0.68` `km, 2.4` `km);`
`vec C = (3.23\cos161^o` `km, 3.23\sin161^o` `km);`
`\qquad = (-3.05` `km, +1.05` `km)`

`vec A + vec B + vec C = (1.22` `km, 2.8` `km)`

Zero Displacement Condition: `vec A + vec B + vec C + vec D = vec 0`
`vec D = - (vec A + vec B + vec C) = (-1.22` `km, -2.80` `km)`

Magnitude: `D=\sqrt{(-1.22km)^2+(-2.80km)^2} = 3.05` `km`.
Orientation: `\theta = 180^o + arctan((-2.80)/(-1.22))`
`\qquad \qquad \qquad \qquad \qquad \quad = 180^o + 66.4^o = 246.4^o`

Note: In calculating the orientation, we recognize that the vector lies in the third quadrant because both the X and Y component are negative. Since the '`\arctan`' function only gives the angle between the vector and the nearest horizontal (negative X-axis), we have to explicitly add `180^o` to make it stick to the convention of measuring all angles counterclockwise to the positive X-axis.