What is a vector in mathematics?

Answer 1

For example, imagine a submarine is at a dangerous depth and we are interested in that fact. You might ask:

What is the temperature down there? In case they freeze down there.

What is the pressure on the hull? In case it is crushed.

What is the speed of the submarine? So they can get out of trouble quickly.

Now, the answers to the first two questions will satisfy your curiosity. They probably will freeze at -12 degrees but the hull can withstand a pressure of 12,000 megapascals, but told that the sub was going at 2 knots wouldn't help. You'd probably ask again, "Up, down, backwards or forwards?"

Pressure and temperature are NOT VECTORS in physics. They just need a quantity to show all about them. VECTORS on the other hand, like velocity, have quantity and also direction, and both of these are necessary to show them. Two knots down is very different from two knots up.

You might argue that the pressure on the top of the sub is down and the pressure on the bottom of the sub is up. This is true, but we know that information without anybody looking or telling us. It's not necessry to be told that. The direction a pressure acts is defined by the situation, but a velocity isn't given by the shape of the object or the direction it's facing.

Therefore, Vectors are qualities with quantity and direction.

For example, suppose you are in a car going at 60 mph due North and there is a cross-wind blowing at 20 mph from the west. What will be the speed and direction of the airflow over the car? If you do that, you're doing a Vector algebra sum.