As discussed earlier, x =
In other words, multiplication by a scalar magnifies the length of the vector. What about the direction? You should convince yourself by drawing a picture that if α is negative, it causes the resulting vector to point in the opposite direction while if α > 0 it preserves the direction the vector points. One way to see this is to first observe that if α≠1, then x and αx are both points on the same line. Note that there is no change in this when you replace ℝ3 with ℝn.