A surveying team fixes the position of a sensor in a tunnel using three…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Tensors and differential geometry

Conditions of Validity, Ch A coordinate system is a map: the metric, the scale factors and where a chart fails

Problem

A surveying team fixes the position of a sensor in a tunnel using three non-orthogonal rods anchored at the origin, giving the basis e1=(1.0,0,0) m\mathbf{e}_1=(1.0,0,0)\ \text{m}, e2=(0,1.0,0) m\mathbf{e}_2=(0,1.0,0)\ \text{m}, e3=(1.0,0,1.0) m\mathbf{e}_3=(1.0,0,1.0)\ \text{m}. ((a)) Construct the reciprocal basis ei\mathbf{e}^i using e1=(e2×e3)/V\mathbf{e}^1=(\mathbf{e}_2\times\mathbf{e}_3)/V, e2=(e3×e1)/V\mathbf{e}^2=(\mathbf{e}_3\times\mathbf{e}_1)/V, e3=(e1×e2)/V\mathbf{e}^3=(\mathbf{e}_1\times\mathbf{e}_2)/V with V=e1⋅(e2×e3)V=\mathbf{e}_1\cdot(\mathbf{e}_2\times\mathbf{e}_3). ((b)) The sensor is at r=(2.0,3.0,5.0) m\mathbf{r}=(2.0,3.0,5.0)\ \text{m}. Write r=ciei\mathbf{r}=c^i\mathbf{e}_i and find c1,c2,c3c^1,c^2,c^3.

Browse the problem library