A particle has velocity in laboratory axes

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

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

A particle has velocity v=(2.0,−3.0,6.0) m s−1\mathbf v = (2.0, -3.0, 6.0)\,\mathrm{m\,s^{-1}} in laboratory axes. A detector defines its own frame with unit vectors e^1=13(1,2,2),e^2=13(2,1,−2),e^3=13(2,−2,1).\hat{\mathbf e}_1 = \tfrac13(1,2,2), \qquad \hat{\mathbf e}_2 = \tfrac13(2,1,-2), \qquad \hat{\mathbf e}_3 = \tfrac13(2,-2,1). ((a)) Verify that {e^1,e^2,e^3}\{\hat{\mathbf e}_1,\hat{\mathbf e}_2,\hat{\mathbf e}_3\} is orthonormal. ((b)) Starting from v=c1e^1+c2e^2+c3e^3\mathbf v = c_1\hat{\mathbf e}_1+c_2\hat{\mathbf e}_2+c_3\hat{\mathbf e}_3 and the orthonormality just shown, derive the formula for each cmc_m and evaluate c1,c2,c3c_1,c_2,c_3 to three significant figures. ((c)) Check the result using c12+c22+c32=∣v∣2c_1^2+c_2^2+c_3^2 = |\mathbf v|^2.

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