How do we know if something is vector or scalar quantity?

[Faiq]

I am well-versed with the definition of scalar and vector quantities.The confusion I mainly have is at many points, my textbook makes ambiguous statements like "Because force is vector quantity it follows that field strength is also a vector quantity."
What relationship should arbitrary quantities X and Y hold, so if X is a vector quantity Y will also be a vector quantity?

 

[Dale]

This is just from the math. A vector times a scalar is another vector. The divergence of a vector is a scalar. The gradient of a scalar is a vector. The dot product of two vectors is a scalar. The cross product of two vectors is a vector (pseudo vector technically). Etc.

 

[anorlunda]

Simply ask yourself if direction is an essential part of the quantity. Sometimes, we use both such as speed (scalar) and velocity (vector) referring to the same thing. It depends on whether the direction is important to you.

Sometimes, we can see it in the signed/unsigned properties.

For example momentum $mv$ is signed + or -, and thus a vector. When two cars collide, it makes a very big difference whether they were traveling in the same direction, or opposite directions.

Kinetic energy $\frac{mv^2}{2}$ is unsigned because $v^2$ is always positive. Thus, K.E. is a scalar. It takes the same fuel energy to accelerate a car to 60 mph eastward as it does to accelerate to 60 mph westward.

 

[Faiq]

Oh so can I say Kinetic energy is scalar because mass(scalar) x v^2 (dot product of vector = scalar ) = KE (scalar) in mathematical terms?

 

[Faiq]

Simply ask yourself if direction is an essential part of the quantity. Sometimes, we use both such as speed (scalar) and velocity (vector) referring to the same thing. It depends on whether the direction is important to you.

Sometimes, we can see it in the signed/unsigned properties.

For example momentum $mv$ is signed + or -, and thus a vector. When two cars collide, it makes a very big difference whether they were traveling in the same direction, or opposite directions.

Kinetic energy $\frac{mv^2}{2}$ is unsigned because $v^2$ is always positive. Thus, K.E. is a scalar. It takes the same fuel energy to accelerate a car to 60 mph eastward as it does to accelerate to 60 mph westward.

Yeah, but we kind of have some intution in this example to determine the nature of the quantity. I was concerned about what if we couldn't use intution to help us arrive at a decent conclusion

 

[Dale]

Oh so can I say Kinetic energy is scalar because mass(scalar) x v^2 (dot product of vector = scalar ) = KE (scalar) in mathematical terms?

Yes, exactly

 

[Stephen Tashi]

What relationship should arbitrary quantities X and Y hold, so if X is a vector quantity Y will also be a vector quantity?

There is no general relationship that makes that implication true for arbitrary quantities.

As others have suggested, one hint about whether Y is a vector quantity is whether it has "direction" as well as "magnitude". However, this is not sufficient. A vector quantity must obey the parallelogram law. Whether Y obeys the parallelogram law depends on how the operation of addition and the operation of scalar multiplication are defined for things of type Y.

The distinction between "scalar" and "vector" is complicated by the fact that one may view scalars as 1-dimensional vectors.