Question 2.7: Magnitude and Direction Cosines of a Vector An engineer desi...

Magnitude and Direction Cosines of a Vector

An engineer designing a threshing machine determines that at a particular time the position vectors of the ends A and B of a shaft are r_{A} = 3i – 4j- 12 K  (ft) and r_{B} = -i + 7j + 6 K  (ft) .

(a) What is the magnitude of r_{A}?
(b) Determine the angles θ_{x} ,  θ_{y} , and θ_{z} . between r_{A} and the positive coordinate
axes.
(c) Determine the scalar components of the position vector of end B of the
shaft relative to end A.

Strategy
(a) Since we know the components of r_{A}, we can use Eq. (2.14) to determine its magnitude.

\left|U\right| = \sqrt{ U^{2}_{x} + U^{2}_{y} + U^{2}_{z}} .     (2.14)

(b) We can obtain the angles θ_{x}, θ_{y} . and θ_{z} , from Eqs.(2.15).

U_{x} = \left|U\right|\cos θ_{x}, U_{y} = \left|U\right|\cos θ_{y}, U_{z} = \left|U\right|\cos θ_{z}.      (2.15)

(c) The position vector of end B of the shaft relative to end A is r_{B}-r_{A}