1.1 Vector definitions and basic operations

Vectors are the building blocks of many physical concepts. They help us describe quantities with both size and direction, like velocity or force. Understanding how to work with vectors is crucial for tackling more complex problems in physics and engineering.

Basic vector operations form the foundation for more advanced vector algebra. Addition, subtraction, and scalar multiplication allow us to combine and manipulate vectors, essential skills for analyzing physical systems and solving real-world problems involving vector quantities.

Vector Fundamentals

Defining Vectors and Scalars

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• Vectors quantities that possess both magnitude and direction
  • Represented by an arrow with length proportional to magnitude and pointing in the direction of the quantity (velocity, force)
• Scalars quantities that have only magnitude, no direction
  • Represented by a single number (temperature, mass, time)
• Magnitude the length or size of a vector, denoted as $||\vec{v}||$ for vector $\vec{v}$
  • Calculated using the Pythagorean theorem in multiple dimensions $||\vec{v}|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}$
• Direction orientation of a vector in space, often given as an angle relative to a reference axis
  • Can be described using unit vectors or spherical/cylindrical coordinates

Unit Vectors

• Unit vectors vectors with a magnitude of 1, used to specify direction
  • Denoted with a hat symbol, e.g., $\hat{i}$, $\hat{j}$, $\hat{k}$ for standard basis vectors in 3D Cartesian coordinates
• Any vector can be expressed as a linear combination of unit vectors scaled by the vector's components
  • For example, $\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$
• Normalizing a vector dividing a vector by its magnitude to create a unit vector pointing in the same direction
  • $\hat{v} = \frac{\vec{v}}{||\vec{v}||}$, where $\hat{v}$ is the unit vector in the direction of $\vec{v}$

Vector Operations

Addition and Subtraction

• Vector addition combining two or more vectors to create a resultant vector
  • Geometrically, performed by placing the tail of one vector at the head of the other and drawing the resultant from the free tail to the free head
  • Component-wise, add corresponding components: $\vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}$
• Vector subtraction finding the difference between two vectors
  • Geometrically, performed by placing the tails of the vectors together and drawing the resultant from the head of the subtrahend to the head of the minuend
  • Component-wise, subtract corresponding components: $\vec{a} - \vec{b} = (a_1 - b_1)\hat{i} + (a_2 - b_2)\hat{j} + (a_3 - b_3)\hat{k}$

Scalar Multiplication

• Scalar multiplication multiplying a vector by a scalar value to change its magnitude and possibly direction
  • Multiplying by a positive scalar changes only the magnitude, while multiplying by a negative scalar reverses the direction
  • Distributive property applies: $c(\vec{a} + \vec{b}) = c\vec{a} + c\vec{b}$, where $c$ is a scalar
• Scaling a vector by a factor $k$ multiplies each component by $k$
  • $k\vec{v} = (kv_1)\hat{i} + (kv_2)\hat{j} + (kv_3)\hat{k}$
• Scalar division dividing a vector by a non-zero scalar value to change its magnitude
  • $\frac{\vec{v}}{c} = (\frac{v_1}{c})\hat{i} + (\frac{v_2}{c})\hat{j} + (\frac{v_3}{c})\hat{k}$, where $c \neq 0$

Vector Representations

Position Vectors

• Position vector a vector that represents the position of a point relative to the origin of a coordinate system
  • Denoted as $\vec{r}$ or $\vec{x}$, with components $(x, y, z)$ in 3D Cartesian coordinates
• Displacement the change in position, calculated as the difference between two position vectors
  • $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$, where $\vec{r}_1$ and $\vec{r}_2$ are the initial and final position vectors
• Velocity and acceleration can be derived from position vectors by taking the first and second derivatives with respect to time
  • Velocity: $\vec{v}(t) = \frac{d\vec{r}}{dt}$, Acceleration: $\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$

Cartesian Components

• Cartesian components the projections of a vector onto the coordinate axes
  • In 2D: $\vec{v} = v_x\hat{i} + v_y\hat{j}$, where $v_x$ and $v_y$ are the x and y components
  • In 3D: $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$, where $v_x$, $v_y$, and $v_z$ are the x, y, and z components
• Component form allows for easy vector arithmetic and analysis
  • Addition, subtraction, and scalar multiplication can be performed component-wise
• Converting between Cartesian components and magnitude-direction form
  • Magnitude: $||\vec{v}|| = \sqrt{v_x^2 + v_y^2 + v_z^2}$
  • Direction angles: $\theta_x = \cos^{-1}(\frac{v_x}{||\vec{v}||}), \theta_y = \cos^{-1}(\frac{v_y}{||\vec{v}||}), \theta_z = \cos^{-1}(\frac{v_z}{||\vec{v}||}$)