Step 1 :Given vectors are: \(\mathbf{u} = \langle 3,-1 \rangle\), \(\mathbf{v} = \langle 2,1 \rangle\), and \(\mathbf{w} = \langle 1,4 \rangle\)
Step 2 :We need to evaluate the expression \(u \cdot v - u \cdot w\)
Step 3 :The dot product of two vectors is calculated by multiplying the corresponding entries of the two vectors and then adding those products
Step 4 :Calculate the dot product of vectors u and v: \(u \cdot v = 3*2 + (-1)*1 = 5\)
Step 5 :Calculate the dot product of vectors u and w: \(u \cdot w = 3*1 + (-1)*4 = -1\)
Step 6 :Subtract the dot product of vectors u and w from the dot product of vectors u and v: \(u \cdot v - u \cdot w = 5 - (-1) = 6\)
Step 7 :Final Answer: \(\boxed{6}\)