Answer to Question #96535 in Geometry for Meaghan

Question #96535
Three non-perpendicular vectors have magnitudes of 242, 57, 159 Newtons and respectively have angles of 24, 67 and 69 degrees with respect to the x-axis. Find the magnitude of their resultant.

Round to two decimal places.
1
Expert's answer
2019-10-17T09:12:49-0400

Let these vectors be named X,Y and Z respectively.

Angle between X and Z vector "(\\theta)=69\\degree-24\\degree=45\\degree"

Magnitude of Resultant vector "(R)=\\sqrt{X^2+Z^2+2XZ cos\\theta}=" "\\sqrt{242^2+159^2+2\\times 242\\times 159 \\times cos 45\\degree}" "=\\sqrt{138252.892}=371.824"

By parallelogram law of vector addition, Angle between resultant vector(R) and X vector"(\\phi)=tan^{-1}(\\frac{Z sin \\theta}{X+Z cos\\theta})=tan^{-1}(0.3172)=17.6\\degree"

Angle between R and Y "(\\delta)=67\\degree-17.6\\degree=49.4\\degree"

Resultant magnitude of R and Y"=\\sqrt{R^2+Y^2+2RY cos\\delta}=" "\\sqrt{371.824^2+57^2+2\\times 371.824\\times 57 \\times cos 49.4 \\degree}" "=410.864835\\approx410.86"

Image for reference(only to assist,not a part of problem):




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