A vector is a quantity described by a magnitude and a direction. The magnitude is always +ve or zero. A -ve sign in front of a vector indicates the same magnitude but in the opposite direction. They are not magnitudes, because they can be +ve or -ve (as you note) depending on the angle between the vector and the axes.

Components of a Vector

A vector is a mathematical object that has both magnitude and direction. In three-dimensional space, a vector can be represented as a combination of its components along the x, y, and z axes. These components describe how much of the vector's magnitude is in each direction.

Option C: Only three components along any of the axes

The correct answer to the given question is option C, which states that a vector can only have three components along any of the axes. Let's understand why this is true.

Three-Dimensional Space

In three-dimensional space, we have three mutually perpendicular axes: x, y, and z. Any vector in this space can be decomposed into its components along each of these axes.

Example of a Vector

Consider a vector V with a magnitude of 5 units and a direction specified by angles α, β, and γ with respect to the x, y, and z axes, respectively. The components of this vector along each axis can be calculated using trigonometry.

The x-component (Vx) can be found using the formula:
Vx = V * cos(α)

The y-component (Vy) can be found using the formula:
Vy = V * cos(β)

The z-component (Vz) can be found using the formula:
Vz = V * cos(γ)

Vector with Three Components

As we can see from the above calculations, a vector in three-dimensional space can have three components along each of the axes. These components represent the magnitude of the vector in each direction.

Unit Vector

A unit vector is a vector that has a magnitude of 1. It is often used to represent a direction. The direction perpendicular to a vector can be represented by a unit vector. However, it is important to note that a vector itself can have components along any axis, including the direction perpendicular to its own direction.

Conclusion

In conclusion, a vector in three-dimensional space can have three components along each of the axes. This allows us to represent the magnitude and direction of the vector accurately. While a vector can have a unit vector along the direction perpendicular to its own direction, it can also have components along that direction. Therefore, option C is the correct answer.

A