91Ó°ÊÓ

We are given the equation \((s+t) \mathbf{u}= s \mathbf{u}+ t \mathbf{u}\), where s and t are scalars, and u is a vector with coordinates (u_x, u_y, u_z). Following the steps in the step-by-step solution, we simplified both sides of the equation: Left Side: \((s + t) \cdot (u_x, u_y, u_z) = ((s + t) \cdot u_x, (s + t) \cdot u_y, (s + t) \cdot u_z)\) Right Side: \(s \cdot (u_x, u_y, u_z) + t \cdot (u_x, u_y, u_z) = (su_x, su_y, su_z) + (tu_x, tu_y, tu_z) = (su_x + tu_x, su_y + tu_y, su_z + tu_z)\) Comparing both sides, we can see that they are equal: \((s+t) \mathbf{u}= s \mathbf{u}+ t \mathbf{u}\). This equation has been proven using the properties of scalar-vector product and vector addition.