Final Answer:
a) The domain of a 3D vector function encompasses all possible input values within three-dimensional space.
b) The range of a 3D vector function comprises all possible output vectors that can be obtained from the function.
c) The magnitude of a 3D vector is the square root of the sum of the squares of its components.
d) The direction of a 3D vector is determined by the angles it makes with the coordinate axes or by normalizing the vector to obtain a unit vector indicating its direction in space.
Step-by-step explanation:
a) Domain of a 3D Vector Function:
In mathematics, the domain refers to the set of all possible input values of a function. In the context of a 3D vector function, the domain encompasses all potential values that the function can take within three-dimensional space. This implies any combination of real numbers for the x, y, and z components of the vectors within the defined coordinate system constitutes the domain of the function.
b) Range of a 3D Vector Function:
The range of a 3D vector function comprises all the possible output vectors that can be obtained by applying the function to the elements within its domain. These output vectors collectively represent the range of the function and essentially form a subset of the three-dimensional space.
c) Magnitude of a 3D Vector:
The magnitude of a 3D vector can be computed using the formula involving the square root of the sum of the squares of its individual components. This calculation results in a scalar value representing the length or magnitude of the vector in space.
d) Direction of a 3D Vector:
Determining the direction of a 3D vector involves understanding the angles it makes with the coordinate axes or normalizing the vector to obtain a unit vector. By normalizing, the vector's length becomes 1, providing insight into its orientation or direction without altering its overall orientation in space. This unit vector signifies the direction the original vector points towards in the 3D coordinate system.