Vectors and Trigonometry: Vector Math and Motion on Ramps

Vectors and Trigonometry in Physics

Scalar and Vector Quantities

Physical quantities in physics are classified as either scalars or vectors. Scalars are described by a single value with units, while vectors require both magnitude and direction for a complete description.

• Scalar Quantities: Examples include length, time, temperature, mass, distance, and speed. Notation: non-bold italic letters (e.g., m, t, d).

• Vector Quantities: Examples include position, displacement, velocity, acceleration, and force. Notation: non-bold italic letters with an arrow above (e.g., $\vec{x}$, $\vec{v}$, $\vec{F}$).

• Vectors are represented in diagrams as arrows, where the arrow's direction indicates the vector's direction and its length is proportional to the magnitude.

Using Vectors

Vector Addition (Graphical Methods)

Vectors can be added graphically using two main methods: the tip-to-tail method and the parallelogram method. The result of vector addition is called the resultant vector.

• Tip-to-Tail Method: Place the tail of the second vector at the tip of the first. The resultant vector points from the tail of the first to the tip of the last vector.

• Order Independence: Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

• Parallelogram Method: Place both vectors tail-to-tail, construct a parallelogram, and the diagonal from the common tail gives the resultant.

Never add only the magnitudes of vectors; direction must always be considered.

Multiplication by a Scalar (Graphical Method)

Multiplying a vector $\vec{A}$ by a scalar $c$ changes its magnitude but not its direction (unless $c$ is negative, in which case the direction reverses).

• If $c > 0$, $c\vec{A}$ points in the same direction as $\vec{A}$ with magnitude $cA$.

• If $c < 0$, $c\vec{A}$ points in the opposite direction with magnitude $|c|A$.

Vector Subtraction (Graphical Method)

The negative of a vector $-\vec{A}$ has the same magnitude as $\vec{A}$ but points in the opposite direction. To subtract vectors, add the negative: $\vec{B} - \vec{A} = \vec{B} + (-\vec{A})$.

Coordinate Systems and Vector Components

Resolving Vectors into Components

For quantitative calculations, vectors are resolved into components along coordinate axes, typically $x$ and $y$. This allows for straightforward addition, subtraction, and scalar multiplication of vectors.

• Any vector $\vec{A}$ in the $xy$-plane can be written as $\vec{A} = \vec{A}_x + \vec{A}_y$.

• The components are found using trigonometry:

$A_x = A \cos \theta$ $A_y = A \sin \theta$

where $A$ is the magnitude and $\theta$ is the angle with respect to the positive $x$-axis.

• If the angle is measured from a different direction, use SOH-CAH-TOA and assign negative signs as needed based on direction.

Vector Operations Using Components

Once vectors are expressed in components, operations become straightforward:

• Addition: $C_x = A_x + B_x$, $C_y = A_y + B_y$

• Subtraction: $C_x = A_x - B_x$, $C_y = A_y - B_y$

• Scalar Multiplication: $C_x = cA_x$, $C_y = cA_y$

The tip-to-tail graphical method is equivalent to adding vectors by their components.

Magnitude and Direction from Components

Given components $A_x$ and $A_y$, the magnitude and direction of $\vec{A}$ are:

$A = \sqrt{A_x^2 + A_y^2}$ $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• If the vector lies in quadrant II or III, add $180^\circ$ to the calculator result for the correct angle.

Motion on a Ramp

Analyzing Motion on Inclined Planes

When analyzing motion on an inclined surface (ramp), it is convenient to use a tilted coordinate system:

• The $x$-axis is aligned along the ramp surface.

• The $y$-axis is perpendicular to the ramp surface.

• Acceleration and force vectors are resolved into components along these axes.

Example: For a sled sliding down a frictionless hill at angle $\theta$, the acceleration along the ramp is $a_x = g \sin \theta$.

Equation Summary

Concept	Equation
Vector components	$A_x = A \cos \theta$, $A_y = A \sin \theta$
Magnitude and direction	$A = \sqrt{A_x^2 + A_y^2}$, $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$
Addition (components)	$C_x = A_x + B_x$, $C_y = A_y + B_y$
Subtraction (components)	$C_x = A_x - B_x$, $C_y = A_y - B_y$
Scalar multiplication	$C_x = cA_x$, $C_y = cA_y$
Motion on a ramp	Use tilted axes: $x$ along ramp, $y$ perpendicular