Vectors and Scalars
Distinguish vector and scalar quantities, resolve vectors into components, and combine vectors graphically and algebraically.
A lesson of Physics, in Measurement, Motion, and Kinematics, about 10 minutes.
By the end of this lesson, you will be able to distinguish scalars from vectors, add vectors graphically and analytically (component method), resolve vectors into components, and apply unit vector notation.
Scalars vs. Vectors
Scalar: a quantity with magnitude only.
Examples: mass (5 kg), temperature (20°C), time (3 s), speed (60 mph), distance (10 m).
Vector: a quantity with both magnitude and direction.
Examples: velocity (60 mph north), displacement (10 m east), force (50 N at 30°), acceleration.
Vector Representation
Graphical: an arrow; length = magnitude; direction = angle.
Component form: $\vec{A} = A_x\hat{i} + A_y\hat{j}$ where $\hat{i}$ and $\hat{j}$ are unit vectors in $x$ and $y$ directions.
Magnitude and direction from components:
$$|\vec{A}| = \sqrt{A_x^2 + A_y^2} \qquad \theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$$
Resolving Vectors into Components
For vector $\vec{A}$ at angle $\theta$ from the positive $x$-axis:
$$A_x = |\vec{A}|\cos\theta \qquad A_y = |\vec{A}|\sin\theta$$
The angle is always measured from positive x-axis (standard position).
Vector Addition
Graphical (tip-to-tail): place vectors tip-to-tail; the resultant goes from start to end.
Component method (preferred for precision):
$$\vec{R} = \vec{A} + \vec{B}: \quad R_x = A_x + B_x, \quad R_y = A_y + B_y$$
Then: $|\vec{R}| = \sqrt{R_x^2 + R_y^2}$ and $\theta = \tan^{-1}(R_y/R_x)$
Example 1 — Adding Two Vectors
$\vec{A} = 30$ m at 0° (east) and $\vec{B} = 40$ m at 90° (north). Find the resultant.
$A_x = 30, A_y = 0$; $B_x = 0, B_y = 40$
$R_x = 30, R_y = 40$
$|\vec{R}| = \sqrt{900+1600} = 50$ m
$\theta = \tan^{-1}(40/30) = 53.1°$ north of east
Example 2 — Resolving at an Angle
A force of 100 N is applied at 37° above the horizontal. Find horizontal and vertical components.
$F_x = 100\cos 37° = 79.9$ N (horizontal)
$F_y = 100\sin 37° = 60.2$ N (vertical)
Example 3 — Subtracting Vectors
$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$
$-\vec{B}$ has the same magnitude but opposite direction. Apply the same component method.
Summary
Scalar: magnitude only. Vector: magnitude + direction.