Scalars and vectors

Physical quantities in physics are classified as either scalars or vectors.

A scalar quantity has only magnitude (size), with no associated direction. Examples include mass, temperature, energy, and time.

A vector quantity has both magnitude and direction. Examples include displacement, velocity, acceleration, and force.

Examples of Scalars and Vectors

Scalars:

• Mass (kg)
• Time (s)
• Temperature (K)
• Distance (m)
• Energy (J)
• Speed (m s$^{-1}$)

Vectors:

• Displacement (m)
• Velocity (m s$^{-1}$)
• Acceleration (m s$^{-2}$)
• Force (N)
• Momentum (kg m s$^{-1}$)

Adding and Subtracting Coplanar Vectors

Vectors can be represented by arrows: the length shows the magnitude, and the arrow points in the direction.

To add vectors:

• Tip-to-tail method: Place the tail of the second vector at the tip of the first. The resultant vector is drawn from the tail of the first to the tip of the last.
• Parallelogram method: Draw both vectors from the same point. Complete the parallelogram; the diagonal represents the resultant vector.

To subtract vectors:

• Reverse the direction of the vector to be subtracted, then add as above.

Resolving a Vector into Components

A vector can be split into two perpendicular components, usually along the horizontal (x) and vertical (y) axes.

If a vector $F$ makes an angle $\theta$ with the horizontal:

• Horizontal component: $F_x = F \cos \theta$
• Vertical component: $F_y = F \sin \theta$

This process is called resolving a vector.

Summary

• Scalars have magnitude only; vectors have magnitude and direction.
• Vectors are added using tip-to-tail or parallelogram methods.
• Any vector can be resolved into perpendicular components using trigonometry.