Understanding Vector Quantities and Projections: Answers to Common Questions

[jlmac2001]

1. Is every quantity that has three components in three dimensions a vector? From the definition of a vector I think it is. If it isn't, can you give me an example?

2. How can you tell if the projection of a force vector F along the velocity vector v of a particle is a scalar?

 

[himanshu121]

It would be possible if it obeys vector law of addition
Consider a current flowing through junction which is attached to three dimension network, it will have magnitude in 3 direction.
But definitely current is a scalar quantity

Projection along v will be fcos(theta)

Now F.v=Fcos(theta)*v Therefore

Fcos(theta)= (F.v)/v which is clearly scalar

 

[M98Ranger]

1. Yes, every quantity that has three components in three dimensions is a vector. This is because a vector is defined as a quantity that has both magnitude and direction. In three dimensions, we need three components (x, y, and z) to fully describe the direction of a vector. An example of a vector with three components could be a force acting on an object with a magnitude of 10 N in the x-direction, 5 N in the y-direction, and 3 N in the z-direction. This vector has both magnitude (10 N) and direction (in the x, y, and z directions), fulfilling the definition of a vector.

2. To determine if the projection of a force vector F along the velocity vector v is a scalar, we can use the dot product. The dot product of two vectors results in a scalar quantity. So, if the dot product of F and v is a scalar, then the projection of F onto v is also a scalar. This means that the force and velocity vectors are perpendicular to each other, resulting in a scalar projection. If the dot product results in a vector, then the projection is not a scalar and the force and velocity vectors are not perpendicular. In this case, the projection would be a vector.