Question

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.

Answer 1

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):