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Multiplication of a vector by a number

The drawing below explains it.

Figure 9

Q. Can you get a vector pointing in some other direction by multiplying a vector by a number?

A. Multiplication by a number can only give you a vector either in the same direction (the number is positive) or a vector in the opposite direction (if the number is negative). Or a zero vector if you multiply by zero. The zero vector has no direction. You can never get a non-zero vector in another direction.

If you had to solve the equation ya = xb , given that b is in a direction different from that of a, the only solution is x = y = 0.

Notice that defining the negative of a vector as a vector of the same length pointing in the opposite direction is consistent with adding the two to get zero: a + (-a) = 0.

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