Yup, prove the sum and difference rules for vector functions. note: a vector fun

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r(t) = f(t)i + g(t)j + h(t)k Let u(t) = x i + y j + z k and v(t) = X i + Y j + Z k SUM Rule: d(u(t)+v(t))/dt = d(x i + y j + z k + X i + Y j + Z k)/dt = (dx/dt + dX/dt)i + (dy/dt + dY/dt)j + (dz/dt + dZ/dt)k d(u(t))/dt = d(x i + y j + z k)/dt = dx/dt i + dy/dt j + dz/dt k d(v(t))/dt = d(X i + Y j + Z k)/dt = dX/dt i + dY/dt j + dZ/dt k d(u(t))/dt + d(v(t))/dt = dx/dt i + dy/dt j + dz/dt k + dX/dt i + dY/dt j + dZ/dt k = d(u(t)+v(t))/dt =>d(u(t)+v(t))/dt = d(u(t))/dt + d(v(t))/dt .......... Hence Sum rule is proved. Difference Rule: d(u(t)-v(t))/dt = d(x i + y j + z k - X i - Y j - Z k)/dt = (dx/dt - dX/dt)i + (dy/dt - dY/dt)j + (dz/dt - dZ/dt)k d(u(t))/dt = d(x i + y j + z k)/dt = dx/dt i + dy/dt j + dz/dt k d(v(t))/dt = d(X i + Y j + Z k)/dt = dX/dt i + dY/dt j + dZ/dt k d(u(t))/dt - d(v(t))/dt = dx/dt i + dy/dt j + dz/dt k - dX/dt i - dY/dt j - dZ/dt k = d(u(t)-v(t))/dt =>d(u(t)-v(t))/dt = d(u(t))/dt - d(v(t))/dt .......... Hence Difference rule is proved.

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