Introduction to Vectors and Vector Components

Vocabulary flashcards defining fundamental vector concepts, trigonometric ratios, unit vectors, and component-based vector addition from the lecture transcript.

Cartesian coordinate system

A coordinate system where the x-axis is parallel to the horizontal and the y-axis is perpendicular to it.

X component

The part or portion of a vector that lies on the horizontal x-axis.

Y component

The part or portion of a vector that lies on the vertical y-axis.

SOHCAHTOA

A trigonometric mnemonic indicating that sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

Sine

A trigonometric function defined as the ratio of the opposite side to the hypotenuse, given by $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

Cosine

A trigonometric function defined as the ratio of the adjacent side to the hypotenuse, given by $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Tangent

A trigonometric function defined as the ratio of the opposite side to the adjacent side, given by $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Unit vectors

Mathematical objects that live on coordinate axes and are used to specify direction in compact mathematical notation.

$$\hat{i}$$

The unit vector pointing along the horizontal x-axis used to specify horizontal direction.

$$\hat{j}$$

The unit vector pointing along the vertical y-axis used to specify vertical direction.

$$\hat{k}$$

The unit vector used in three-dimensional coordinate systems to specify direction along the z-axis.

Unit vector notation

A method of expressing a vector as a linear combination of its scalar components and unit vectors, such as $\mathbf{a} = a_x\hat{i} + a_y\hat{j}$.

Vector addition by components

A method of adding two or more vectors by grouping and summing their like components along each coordinate axis.