Vectors in Physics Cheat Sheet

Key Facts

• A vector has magnitude and direction, while a scalar has magnitude only, such as mass, time, or temperature.
• For a vector $\vec{A}$ at angle $\theta$ from the positive $x$-axis, the components are $A_x = A\cos\theta$ and $A_y = A\sin\theta$.
• The magnitude of a two-dimensional vector is $A = \sqrt{A_x^2 + A_y^2}$.
• The direction angle of a vector can be found with $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$, but the quadrant must be checked.
• Vector addition by components uses $R_x = A_x + B_x$ and $R_y = A_y + B_y$, then $R = \sqrt{R_x^2 + R_y^2}$.
• Unit vector notation writes a vector as $\vec{A} = A_x\hat{i} + A_y\hat{j}$ in two dimensions.
• The dot product is $\vec{A}\cdot\vec{B} = AB\cos\theta = A_xB_x + A_yB_y$, where $\theta$ is the angle between the vectors.
• For projectile motion without air resistance, horizontal and vertical components are independent, so $v_x$ stays constant while $v_y$ changes due to $a_y = -g$.

Vocabulary

Vector

A quantity that has both magnitude and direction, such as displacement, velocity, acceleration, or force.

Scalar

A quantity that has magnitude only and no direction, such as speed, distance, mass, or time.

Component

One part of a vector along a chosen axis, such as $A_x$ along the $x$-axis or $A_y$ along the $y$-axis.

Resultant

The single vector that has the same effect as two or more vectors added together.

Unit Vector

A vector with magnitude $1$ that shows direction, commonly written as $\hat{i}$, $\hat{j}$, or $\hat{k}$.

Dot Product

An operation that multiplies two vectors to produce a scalar using $\vec{A}\cdot\vec{B} = AB\cos\theta$.