Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined.
In four dimensions, angular momentum is no longer represented by a vector. The position is a vector, x μ.The momentum is a vector, p ν.But when you take the cross product you get a rank two tensor, J μν = x μ p ν - p μ x ν. Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension.
Energy is a scalar quantity.Answer2: It depends on the angle!Energy can be a scalar or a vector; consider the vectors F forceand D displacement:FD = -F.D + FxD = FD (cos(angle) + v sin(angle)).F.D is called work a form of energy and is a scalar; FxD iscalled Torque and is a vector form of energy. Both work and Torquehave units of joules or newton times meters.Energy like many quantities in physics is a quaternionconsisting of a scalar part and a vector part; E = Escalar +Evector = E(cos(angle) + v sin(angle)), whether the quantityis a scalar or a vector or both depends on the angle.
Suppose you know that a ball is rolling on a flat table at 15 degrees from a direction parallel to the bottom edge at a speed of 7.0 meters/second. You may want to find out how long the ball will take to roll off the edge 1.0 meter to the right.Define your axes so the ball is at the origin initially and the x-axis is parallel to the bottom edge of the table (refer to the figure). Therefore, the problem breaks down to finding out how long the ball will take to roll 1.0 meter in the x direction. To find the time, you first need to know how fast the ball is moving in the x direction.The problem tells you that the ball is rolling at a speed of 7.0 meters/second at 15 degrees to the horizontal (along the positive x-axis), which is a vector: 7.0 meters/second at 15 degrees gives you both a magnitude and a direction. What you have here is a velocity — the vector version of speed.
The ball’s speed is the magnitude of its velocity vector, and when you include a direction to that speed, you get the velocity vector v.To find out how fast the ball is traveling toward the table edge, you need not the ball’s total speed but the x component of the ball’s velocity. The x component is a scalar (a number, not a vector), and you write it like this: v x.
The y component of the ball’s velocity vector is v y. Therefore, you can say that. The magnitude of a vector v is expressed as v, and from the figure, you can see that the following is true:The two vector-component equations are worth knowing because you see them a lot in any beginning physics course. Make sure you know how they work, and always have them at your fingertips.Of course, if you forget these equations, you can always retrieve them from basic trigonometry.
You may remember that the sine and cosine of an angle in a right triangle are defined as the ratio of the opposite side and the adjacent side to the hypotenuse, like so. You know thatso you can find the x component of the ball’s velocity, v x, this way:Plugging in the numbers gives youYou now know that the ball is traveling at 6.8 meters/second to the right.
Momentum Vector Diagram
Angular Momentum Vector
And because you also know that the table’s edge is 1.0 meter away, you can divide distance by speed to get the time:Because you know how fast the ball is going in the x direction, you now know the answer to the problem: The ball will take 0.15 seconds to fall off the edge of the table. What about the y component of the velocity? That’s easy to find, too.