Vector Addition and Scalar Basics

This flashcard set covers the fundamental definitions and formulas for scalar and vector quantities, as well as the graphical and analytical methods of vector addition.

Head-to-Tail Method

A graphical method used for two or more vectors where the tail of the second vector is placed at the head of the first, and the resultant is drawn from the tail of the first to the head of the last.

Parallelogram Method

A graphical method used for two vectors only, where both vectors are drawn from a common starting point and the resultant is the diagonal of the completed parallelogram.

Triangle Method

A graphical method for two vectors that follows the same steps as the head-to-tail method for two vectors.

Resultant

The vector representing the sum of two or more vectors, denoted as $R$.

Scalar

A quantity that has magnitude only and no direction; examples include speed, distance, time, mass, temperature, and energy.

Vector

A quantity that has both magnitude and direction; examples include force, displacement, velocity, acceleration, weight, and momentum.

X-Component ($R_x$)

The horizontal part of a vector resolved using the formula $R \times \text{cos}(\theta)$, where $\theta$ is measured from the positive x-axis.

Y-Component ($R_y$)

The vertical part of a vector resolved using the formula $R \times \text{sin}(\theta)$, where $\theta$ is measured from the positive x-axis.

Magnitude of the Resultant

The total size of the combined vectors calculated via the Pythagorean theorem: R = \text{\sqrt{R_x^2 + R_y^2}}.

Direction of the Resultant ($\theta$)

The angle calculated using the formula $\theta = \text{tan}^{-1}(\frac{R_y}{R_x})$, always checking the signs of the components for the correct quadrant.

East

A cardinal direction associated with the positive x-axis ($+x$).

West

A cardinal direction associated with the negative x-axis ($-x$).

North

A cardinal direction associated with the positive y-axis ($+y$).

South

A cardinal direction associated with the negative y-axis ($-y$).

Quadrant II

The quadrant where the x-component is negative ($-$) and the y-component is positive ($+$).

Quadrant III

The quadrant where both the x-component and the y-component are negative ($-$).

Quadrant IV

The quadrant where the x-component is positive ($+$) and the y-component is negative ($-$).

Scale

A ratio used to draw vectors accurately on paper, such as $1\, \text{cm} = 10\, \text{N}$.