Vectors and Physical Quantities Lecture Notes

Flashcards covering the definitions, types, and mathematical laws of vectors based on the physics lecture notes.

Physical Quantities

Any property that can be measured or quantified.

Pure Ratio

A type of physical quantity that has a value but no defined units and no direction (e.g., Mechanical Advantage, $\text{U}$).

Scalar

A physical quantity that has a value and a unit, but no defined direction (e.g., mass, energy).

Vector

A physical quantity having a magnitude, unit, direction, and following the triangle or parallelogram rule of vector addition (e.g., displacement).

Null / Zero Vector

A vector having zero magnitude and an undefined direction.

Parallel Vectors

Vectors pointing in the same direction.

Anti-Parallel Vectors

Vectors pointing in opposite directions.

Equal Vectors

Vectors pointing in the same direction with equal magnitude.

Negative of a Vector

Anti-parallel vectors with equal magnitude, represented as $\bar{A} = -\bar{B}$.

Unit Vector ($\hat{n}$)

A vector having unit magnitude ($1\,unit$) that represents only information of direction.

Angle Between 2 Vectors

The angle formed when vectors are arranged tail to tail.

Triangle Law of Addition

A method where the tail of the second vector is attached to the head of the first; the resultant vector is drawn from the tail of the first to the head of the second.

Resultant Magnitude Formula

The formula $C = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$, where $A$ and $B$ are magnitudes and $\theta$ is the angle between them.

Direction of Resultant ($\alpha$)

Calculated using the formula $\tan(\alpha) = \frac{B \sin(\theta)}{A + B \cos(\theta)}$.

Polygon Law of Vector Addition

If the tail of each vector is added to the head of the previous vector, the resultant is given by closing the polygon.

Parallelogram Law of Vector Addition

If two vectors are attached tail to tail and a parallelogram is completed, the resultant vector is the diagonal from the common tail point.

Dot Product (Scalar Product)

The product of two vectors where $\vec{A} \cdot \vec{B} = AB \cos(\theta)$. In Cartesian form: $a_x b_x + a_y b_y + a_z b_z$.

Cross Product (Vector Product)

A product of two vectors that produces a vector output with magnitude $|\vec{A} \times \vec{B}| = AB \sin(\theta)$ and direction determined by the right-hand thumb/screw rule.

Position Vector

A vector representing the direction of any object with respect to an observer or origin.

Displacement

The change in position, represented as $\Delta \vec{r} = \vec{r}_2 - \vec{r}_1$.

Torque ($\vec{\tau}$)

The moment of force, defined as the cross product $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector and $\vec{F}$ is the applied force.