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?

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 the fourth side D.

enter image source here

A: What is the length of the vector D in kilometers?

B: What is the orientation of the vector D, in degrees W of S?

1 Answer
Nov 16, 2017

Answer:

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#

Explanation:

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.