Final answer:
To find -v and 3v for a vector ν = (9, -6), reverse the direction of the vector for -v resulting in (-9, 6), and scale the vector by 3 for 3v, resulting in (27, -18). Vector subtraction is the addition of a negative vector, which does not change based on the order of subtraction.
Step-by-step explanation:
The question involves understanding the concept of vector algebra. When you have a vector ν = (9, -6), the negative of that vector, denoted as -ν, simply reverses the direction of the vector, meaning that -ν = (-9, 6). Multiplying a vector by a scalar, like 3ν, scales the magnitude of the vector by that scalar without changing its direction, so 3ν = (27, -18).
To perform vector subtraction and understand its properties, we learn that subtracting vector B from vector A is akin to adding the negative of B to A. This means that A - B is the same as A + (-B). The order of subtraction doesn't affect the result, which is true for scalars as well as vectors.
When considering the subtraction in an applied context, such as navigation, subtracting vector B from vector A represents a reversal in the direction one should travel when following a path. The magnitude of the vector remains the same, but the direction is opposite to the original vector B.