Vectors: Concepts, Operations, and Applications in Physics today

Vectors in Physics- Smart 24th

Review of Vectors and Scalars

In physics, quantities are classified as either vectors or scalars. This distinction is fundamental for describing motion and forces.

• Scalar: A quantity with only magnitude (size). Examples: temperature, distance, speed.

• Vector: A quantity with both magnitude and direction. Examples: force, displacement, velocity.

Measurement	Quantity	Magnitude?	Direction?	Vector/Scalar
"It's 67°F outside"	Temperature	Yes	No	Scalar
"I pushed with 100 N west"	Force	Yes	Yes	Vector
"I walked for 10 miles"	Distance	Yes	No	Scalar
"I walked 10 mi east"	Displacement	Yes	Yes	Vector
"I drove at 40 mph"	Speed	Yes	No	Scalar
"I drove 60 mph west"	Velocity	Yes	Yes	Vector

Intro to Vector Math

Vectors are represented by arrows and require special rules for addition and subtraction due to their directional nature.

• Combining Scalars: Simple addition (e.g., 3 kg + 4 kg = 7 kg).

• Combining Parallel Vectors: Add like normal numbers if vectors are in the same direction.

• Combining Perpendicular Vectors: Use the Pythagorean theorem (triangle math): $| ext{Resultant}| = ext{sqrt}(A^2 + B^2)$

Example:

• If you walk 5 m to the right, then 5 m up, your total displacement is $| ext{Resultant}| = ext{sqrt}(5^2 + 5^2) = 7.07$ m.

Adding Vectors Graphically

Vectors are added by placing them tip-to-tail. The resultant vector is the shortest path from the start of the first to the end of the last.

• Order does not matter when adding vectors.

• For perpendicular vectors: $| ext{Resultant}| = ext{sqrt}(A^2 + B^2)$

• For any vectors: Use graphical or component methods.

Example:

• Find the magnitude of $extbf{C} = extbf{A} + extbf{B}$ where $extbf{A}$ and $extbf{B}$ are given graphically.

Subtracting Vectors Graphically

Subtracting vectors is similar to addition, but you reverse the direction of the vector being subtracted.

• Resultant vector is the shortest path from the start of the first to the end of the last.

• "Negative" vector: same magnitude, opposite direction.

Example:

• Find the magnitude of $extbf{D} = extbf{C} - extbf{A}$.

Adding Multiples of Vectors

Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative).

• Multiplying by $>1$ increases magnitude.

• Multiplying by $<1$ decreases magnitude.

• Multiplying by $-1$ reverses direction.

Example:

• Find the magnitude of $extbf{R} = 3 extbf{A} - 2 extbf{B}$.

Vector Composition and Decomposition

Vectors can be broken down into components along the x and y axes using trigonometry.

• Composition: Combine components to form a vector.

• Decomposition: Break a vector into its x and y components.

• Use SOH-CAH-TOA: $A_x = A ext{cos}( heta)$ $A_y = A ext{sin}( heta)$

Example:

• Given $A_y = 12$ m at $67^ ext{o}$, find $A$ and $A_x$.

Vector Addition by Components

To add vectors, sum their respective components:

• $R_x = A_x + B_x$

• $R_y = A_y + B_y$

• Magnitude: $| extbf{R}| = ext{sqrt}(R_x^2 + R_y^2)$

• Direction: $heta = an^{-1}(R_y / R_x)$

Example:

• Calculate the magnitude and direction of $extbf{W} = extbf{A} - 2 extbf{B}$.

Math with Vectors in Any Quadrant

Vectors can point in any direction, so their components may be positive or negative depending on the quadrant.

• Adjust signs of components based on direction.

• Absolute angle: $heta = an^{-1}(|A_y / A_x|)$

Example:

• Calculate $A_x$, $A_y$, and the absolute angle for a given vector.

Describing Directions with Words

Directions may be described using compass angles or relative to axes.

• CW/CCW from axis: e.g., $30^ ext{o}$ south of east.

• Use trigonometry to find components.

Example:

• Draw each vector and calculate its components for given compass directions.

Unit Vectors

Unit vectors are used to specify direction and have a magnitude of 1.

• Notation: $extbf{A} = A_x extbf{i} + A_y extbf{j}$

• $extbf{i}$ points in the x-direction, $extbf{j}$ in the y-direction.

Example:

• Express $extbf{A} = 4 extbf{i} + 2 extbf{j}$ and $extbf{B} = -1 extbf{i} + 2 extbf{j}$ in unit vector form.

Dot Product (Scalar Product)

The dot product of two vectors produces a scalar and measures how much one vector extends in the direction of another.

• Formula: $extbf{A} ullet extbf{B} = | extbf{A}| | extbf{B}| ext{cos}( heta)$

• Component form: $extbf{A} ullet extbf{B} = A_x B_x + A_y B_y$

Example:

• Calculate $extbf{A} ullet extbf{B}$ for given vectors.

Cross Product (Vector Product) and the Right-Hand Rule

The cross product of two vectors produces a vector perpendicular to both, with magnitude given by:

• Formula: $| extbf{A} imes extbf{B}| = | extbf{A}| | extbf{B}| ext{sin}( heta)$

• Direction: Use the right-hand rule.

• Component form: $extbf{A} imes extbf{B} = (A_y B_z - A_z B_y) extbf{i} + (A_z B_x - A_x B_z) extbf{j} + (A_x B_y - A_y B_x) extbf{k}$

Example:

• Find the magnitude and direction of $extbf{C} = 2 extbf{i} imes extbf{j}$.

Summary Table: Vector Operations

Operation	Result	Formula
Addition	Vector	$extbf{A} + extbf{B}$
Subtraction	Vector	$extbf{A} - extbf{B}$
Dot Product	Scalar	$extbf{A} ullet extbf{B} = | extbf{A}| | extbf{B}| ext{cos}( heta)$
Cross Product	Vector	$| extbf{A} imes extbf{B}| = | extbf{A}| | extbf{B}| ext{sin}( heta)$

Applications and Examples

• Displacement, velocity, and force are all vector quantities and require vector math for correct analysis.

• Dot and cross products are used in work, torque, and angular momentum calculations.

Practice Problems:

• Calculate the resultant of multiple vectors using graphical and component methods.

• Find the dot and cross products for given vectors and interpret their physical meaning.

Additional info: These notes cover the essential mathematical tools for describing motion and forces in physics, as outlined in Chapter 3: Kinematics in 2D & Vectors.