Vector Fundamentals and Operations Notes

Vector fundamentals

• In physics, a vector is an object that has both direction and magnitude, often shown as arrows. An arrow represents a vector pointing from the origin, with length equal to its magnitude and direction given by the angle from the x-axis. This is related to polar coordinates, where you might describe a vector by its radial distance $r$ and angle $\theta$, but we will describe vectors primarily by components.
• A 2D vector is written as $\mathbf{v}=(v<em>x,\,v</em>y)$ where $v<em>x$ is the x-component and $v</em>y$ is the y-component. The arrow over a letter (e.g., $\vec{v}$ or $\mathbf{v}$) denotes that the quantity is a vector, not a simple coordinate pair.
• Examples of vectors in components:
  • If the vector is $\mathbf{v}=(3,\,2)$ then you go 3 units in +x and 2 units in +y from the origin.
  • If the vector is $\mathbf{u}=(1,4)$, drawn from the origin, it goes 1 in x and 4 in y.
• The magnitude (length) of a vector is the length of its arrow, and the direction is the orientation of the arrow (the angle from the x-axis).
• Distinctions in physics:
  • Vectors vs scalars: vectors have both magnitude and direction; scalars have magnitude only.
  • Examples: velocity, displacement, and acceleration are vectors; time, mass, and temperature are scalars.
• Velocity is a vector; time is a scalar. Displacement is a vector describing change in position; distance is the magnitude of displacement (a scalar).
• Velocity and acceleration definitions in vector form:
  • Velocity can be written as the change in displacement over time: $\mathbf{v}=\frac{\Delta \mathbf{r}}{\Delta t}$ or, in continuous form, $\mathbf{v}=\frac{d\mathbf{r}}{dt}$.
  • Acceleration is the change in velocity over time: $\mathbf{a}=\frac{\Delta \mathbf{v}}{\Delta t}=\frac{d\mathbf{v}}{dt}$.
• Displacement is the difference between final and initial positions: $\Delta \mathbf{r}=\mathbf{r}<em>f-\mathbf{r}</em>i\,.$ Distance is the magnitude of that displacement: $\text{distance}=|\Delta \mathbf{r}|$.
• Time is a scalar: it does not have a direction. Expressions with time use scalar arithmetic.
• Mass and temperature are scalars (e.g., 90 kg, 50 °C) and do not carry directional information.
• When thinking about whether something is a vector, ask: does it have a direction? If yes, it’s a vector; if not, it’s a scalar.
• In this course, the primary focus is on velocity, displacement, and acceleration as vectors, and on using scalars like time, mass, and temperature separately.
• Conceptual takeaway: any math in physics is done with vectors and component-wise operations (i.e., break into x and y components and perform arithmetic on components).

Basic vector operations

• Scalar multiplication
  • If you multiply a vector by a scalar $c$, the result is the vector scaled along its direction: $c\,\mathbf{v}=(c\,v<em>x,\,c\,v</em>y)$.
  • Examples:
  • If $\mathbf{a}=(1,4)$, then $3\mathbf{a}=(3,12)$.
  • If you multiply by a negative scalar, the vector flips direction: $-1\cdot\mathbf{a}=(-v<em>x,-v</em>y)$.
  • Multiples of a vector all lie on the same line through the origin.
• Vector addition
  • For vectors $\mathbf{u}=(u<em>x, u</em>y)$ and $\mathbf{v}=(v<em>x, v</em>y)$, their sum is $\mathbf{u}+\mathbf{v}=(u<em>x+v</em>x,\,u<em>y+v</em>y)$.
  • Example: $(1,4)+(3,2)=(4,6)$.
  • Geometrically, you can add vectors using the tip-to-tail method or the parallelogram rule: placing the tail of one vector at the tip of the other yields the resultant vector as the diagonal of the parallelogram.
• Subtraction
  • Subtraction is vector addition with a negative: $\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v})$.
  • Example: $(1,4) - (3,2) = (-2,2)$.
  • The opposite vector (negative) points in the exact opposite direction; subtracting changes signs of the components.
• Important caveat about multiplication
  • There is no general product of two vectors that yields a vector in the same simple sense. The transcript emphasizes that you should not expect to multiply two vectors in the same way you multiply scalars; there are specific products (like dot or cross products) but not a universal vector multiplication rule.
• Practical note for physics problem-solving
  • Always work with vector components (x and y) and perform component-wise arithmetic.
  • When summing or subtracting, track both components; the resultant vector is formed by adding/subtracting corresponding components.

Geometric interpretation and visualization

• Tip-to-tail visualization (parallelogram rule) helps you see how addition works graphically.
• Subtraction corresponds to adding the opposite vector, which flips direction.
• Graphical intuition is useful for understanding direction and how the resultant vector behaves in planes like the x-y plane.
• In physics problems, the pictures are often approximations; the algebra on components is what gives you precise results.

Velocity, displacement, and acceleration: physics context

• Vector vs scalar in a physics equation:
  • Velocity, displacement, and acceleration are vectors (have arrows over them).
  • Time is a scalar (no arrow).
  • Mass and temperature are scalars.
• Core relationships:
  • Velocity is the rate of change of displacement: $\mathbf{v}=\frac{d\mathbf{r}}{dt}$ and, for discrete changes, $\mathbf{v}=\frac{\Delta \mathbf{r}}{\Delta t}$.
  • Acceleration is the rate of change of velocity: $\mathbf{a}=\frac{d\mathbf{v}}{dt}$ and, for discrete changes, $\mathbf{a}=\frac{\Delta \mathbf{v}}{\Delta t}$.
• Signs and directions matter: components can be positive or negative, indicating direction along each axis.
• Common scalar examples (for contrast): time $t$, mass $m$, temperature (e.g., 50 °C) are scalars; they do not specify a direction.
• When analyzing equations, identify which quantities are vectors and perform operations component-wise.

Example problem: acceleration from velocity data

• Given a velocity final $\mathbf{v}<em>f=(1,5)$, a velocity initial $\mathbf{v}</em>i=(2,6)$, and a time interval $t=67$, find the acceleration $\mathbf{a}$.
• Start from the velocity-time relation: $\mathbf{v}<em>f = \mathbf{v}</em>i + \mathbf{a} t$.
• Solve for the acceleration: subtract the initial velocity and divide by time:
  • $\mathbf{a} = \frac{\mathbf{v}<em>f - \mathbf{v}</em>i}{t} = \frac{(1,5)-(2,6)}{67} = \left(-\frac{1}{67},\,-\frac{1}{67}\right).$
• Magnitude of acceleration:
  • $|\mathbf{a}| = \sqrt{\left(-\frac{1}{67}\right)^2 + \left(-\frac{1}{67}\right)^2} = \sqrt{\left(\frac{1}{67^2}\right) + \left(\frac{1}{67^2}\right)} = \sqrt{\frac{2}{67^2}} = \frac{\sqrt{2}}{67} \approx 0.0211.$
• Direction of acceleration:
  • Since both components are negative and equal, the vector lies along the line $y=x$ in the third quadrant. The angle from the +x axis is
  • $\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$ (or equivalently, $-135^{\circ}$).
• Note on the transcript’s numeric value: the speaker stated the magnitude as approximately $1.113\times 10^{-4}$, which is inconsistent with the computed magnitude above. The correct magnitude is about $|\mathbf{a}| \approx 0.0211.$ The exact symbolic form is $|\mathbf{a}| = \dfrac{\sqrt{2}}{67}.$
• Summary: the acceleration vector is $\mathbf{a}=\left(-\frac{1}{67},\,-\frac{1}{67}\right)$ with magnitude $|\mathbf{a}|=\dfrac{\sqrt{2}}{67} \approx 0.0211$ and direction in the third quadrant (angle $\frac{5\pi}{4}$ from +x or -135°).
• The absolute length notation for magnitude is often shown with double vertical lines around a number or vector; symbolically, the magnitude is a scalar value (the length of the arrow).
• Final practical point: break vectors into components and use component-wise arithmetic to solve problems; this helps you determine both magnitude and direction of the resultant vector.

Quick recap: key distinctions and rules

• Vector vs scalar:
  • Vectors have magnitude and direction; scalars have magnitude only.
  • Examples: $\text{vector}$: $\mathbf{v}, \mathbf{r}, \mathbf{a}$; $\text{scalar}$: $t, m, T$.
• Operations on vectors:
  • Scalar multiplication: $c\,\mathbf{v}=(c\,v<em>x, c\,v</em>y)$.
  • Vector addition: $\mathbf{u}+\mathbf{v}=(u<em>x+v</em>x, u<em>y+v</em>y)$.
  • Subtraction: $\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v}) = (u<em>x-v</em>x, u<em>y-v</em>y)$.
  • No general vector multiplication (there are specific products like dot and cross products, but no universal vector multiplication rule).
• Practical habit: always use components for algebra in physics problems; this keeps direction and magnitude clear and helps with subsequent steps like rotations or decompositions.
• In physics problems, be mindful of units (e.g., meters per second for velocity, meters per second squared for acceleration); magnitudes often come with units, and mixing units can lead to mistakes.
• Course logistics mentioned in the transcript (not concepts): daily notes, link to a PDF with course notes, and posting homework problems from the textbook on the course homepage under Resources.