Scalars vs Vectors Cheat Sheet

Key Facts

• A scalar quantity has magnitude only, such as mass, time, temperature, speed, energy, or distance.
• A vector quantity has magnitude and direction, such as displacement, velocity, acceleration, force, or momentum.
• The magnitude of a two-dimensional vector is $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$.
• The horizontal component of a vector is $A_x = A\cos\theta$ when $\theta$ is measured from the positive $x$-axis.
• The vertical component of a vector is $A_y = A\sin\theta$ when $\theta$ is measured from the positive $x$-axis.
• The resultant of two perpendicular vectors has magnitude $R = \sqrt{A^2 + B^2}$.
• The direction of a vector from its components can be found with $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$, with the quadrant checked separately.
• Vectors in the same direction add, while vectors in opposite directions subtract, so $\vec{R} = \vec{A} + \vec{B}$ depends on direction.

Vocabulary

Scalar

A scalar is a quantity described by magnitude only and no direction.

Vector

A vector is a quantity described by both magnitude and direction.

Magnitude

Magnitude is the size or amount of a scalar or vector quantity.

Direction

Direction tells which way a vector points, often measured as an angle or described with compass directions.

Component

A component is one part of a vector along a chosen axis, such as $A_x$ or $A_y$.

Resultant

The resultant is the single vector that has the same effect as two or more vectors combined.