Chapter 2 Intro To Vectors


Scalar and Vector Quantities

  • Scalar Quantities:

    • Defined as physical quantities that are completely described by a single numerical value (magnitude) alone.

    • Contain no associated direction.

    • Examples of scalar quantities include:

    • Number of people in a room

    • Temperature

    • Mass

    • Speed

    • Distance

    • Mathematical operations on scalar quantities follow standard single-number arithmetic and algebra learned since grade school.

  • Vector Quantities:

    • Defined as physical quantities described by both a magnitude (how much or the quantity size) and a direction.

    • Examples of vector quantities include:

    • Force: Defined fundamentally as a push or a pull, inherently requiring a directional orientation (e.g., toward, away, right, left, up, down).

    • Velocity: Describes both how fast an object is moving (speed) and its direction of motion, conveying complete kinematical information.

    • Displacement: Measures the change in position of an object, requiring both distance moved and direction relative to an origin.

    • Acceleration: Specifies the rate of change of velocity per unit time in a specific direction.

    • Momentum: Product of mass and velocity, maintaining vector directionality.

    • Torque: Specifies rotational force applied in a given directional orientation.

Graphical Representation and Measurement of Vectors

  • Arrow Representation:

    • A vector is visually depicted as an arrow on a coordinate plane.

    • Tail: The starting point of the vector arrow.

    • Tip (or Head): The pointed end of the vector arrow indicating direction.

    • Length: Scaled proportionally to represent the magnitude of the physical vector quantity.

    • Orientation: Represents the directional angle relative to a defined reference axis.

  • Scaling Vector Magnitudes:

    • Non-length vector quantities (e.g., force, velocity, acceleration) are mapped onto geometric lengths using a scaling factor.

    • Example scale factor: 1\n,text{cm} = 1\n,text{N} or 1\n,text{cm} = 10\n,text{m/s}.

    • Units of velocity magnitude in Standard International (SI) units are meters per second (\n,text{m/s}), and in US customary units are miles per hour (\n,text{mph}).

  • Measurement Tools and Procedure:

    • Ruler: Used to measure the linear magnitude (length) of the vector arrow.

    • Protractor: Used to measure the angle of orientation.

    • Step-by-Step Measurement Procedure:

    • Place the center crosshair/origin mark of the protractor precisely at the tail of the vector.

    • Align the horizontal baseline (0∘0^\circ) of the protractor with the positive x-axis.

    • Align the vertical baseline (90∘90^\circ) with the positive y-axis.

    • Place the 0\n,text{cm} mark of a ruler at the tail of the vector along the arrow shaft.

    • Read the tip location on the ruler (e.g., 12.6\n,text{cm} with a millimeter tick precision yielding an uncertainty of \pm 1\n,text{mm} or \pm 0.1\n,text{cm}).

    • Convert the length to physical units using the scale factor: 12.6\n,text{cm} \times \left(\frac{1\n,text{N}}{1\n,text{cm}}\right) = 12.6\n,text{N}.

Polar Form and Vector Equality

  • Polar Representation:

    • Denoted symbolically by a letter with an arrow overhead (e.g., a⃗\vec{a}), or printed in boldface (e.g., a\mathbf{a}).

    • Magnitude is expressed using absolute value brackets around the vector symbol: ∣a⃗∣|\vec{a}|.

    • In one-dimensional motion, direction is indicated by positive (++) or negative (−-) signs relative to a reference point, and ∣a⃗∣|\vec{a}| strips the sign to give magnitude.

    • In polar form, a vector is explicitly defined by its magnitude and angle relative to a reference axis:     \vec{a} = 12.6\n,text{N} \text{ at } 24^\circ \text{ counterclockwise from the positive x-axis}

    • Shorthand polar notation: (|\vec{a}|, \theta) = (12.6\n,text{N}, 24^\circ).

  • Equality of Vectors:

    • Two vectors a⃗\vec{a} and b⃗\vec{b} are defined as equal (a⃗=b⃗\vec{a} = \vec{b}) if and only if they possess identical magnitudes and point in the exact same direction.

    • Spatial translation invariance: A vector can be moved/translated to any position in space without changing its identity, provided its magnitude and orientation angle are preserved.

    • Vectors do not need to originate from the same starting location to be equal.

  • Vector Comparison Case Study:

    • a⃗\vec{a} points from (0,0)(0,0) to (1,1)(1,1).

    • b⃗\vec{b} points from (2,3)(2,3) to (3,4)(3,4).

    • c⃗\vec{c} points from (3,3)(3,3) to (2,2)(2,2).

    • Analysis: a⃗\vec{a} and b⃗\vec{b} both undergo a displacement of +1+1 unit horizontally and +1+1 unit vertically, giving them equal magnitudes and direction angles (a⃗=b⃗\vec{a} = \vec{b}).

    • Vector c⃗\vec{c} undergoes a displacement of −1-1 unit horizontally and −1-1 unit vertically; it has the exact same magnitude as a⃗\vec{a} and b⃗\vec{b}, but points in the exact opposite direction (c⃗=−a⃗=−b⃗\vec{c} = -\vec{a} = -\vec{b}).

Coordinate Systems, Angle Conventions, and Radians

  • Standard Angle Convention:

    • Unless explicitly stated otherwise, vector direction angles θ\theta are measured counterclockwise (CCW) using the positive x-axis as the primary reference line (0∘0^\circ).

    • Four Quadrants of the 2D Cartesian Plane:

    • Quadrant I: 0^\circ < \theta < 90^\circ

    • Quadrant II: 90^\circ < \theta < 180^\circ

    • Quadrant III: 180^\circ < \theta < 270^\circ

    • Quadrant IV: 270^\circ < \theta < 360^\circ

  • Negative Vectors and Angles:

    • Vector magnitudes are strictly non-negative quantities (∣a⃗∣≥0|\vec{a}| \ge 0).

    • Multiplying a vector by a negative scalar (−1-1) reverses its direction by 180∘180^\circ.

    • In polar form, −a⃗-\vec{a} is represented by keeping the magnitude ∣a⃗∣|\vec{a}| positive and modifying the direction angle to θ+180∘\theta + 180^\circ.

  • Degrees and Radians:

    • Degree: A unit of angular measure where a full rotation comprises 360∘360^\circ.

    • Radian: A dimensionless unit of angular measure defined as the ratio of arc length (ss) to radius (rr):     θ=sr\theta = \frac{s}{r}

    • A full circle contains 2\pi\n,text{rad} \approx 6.28318\n,text{rad}, equivalent to 360∘360^\circ.

    • \pi\n,text{rad} = 180^\circ.

    • One Radian: The subtended angle when arc length equals radius (s=rs = r). 1\n,text{rad} = \frac{180^\circ}{\pi} \approx 57.2958^\circ (roughly 54∘−57∘54^\circ - 57^\circ).

    • Conversion equations:

    • Degrees to radians: \theta_{\n,text{rad}} = \theta_{\n,text{deg}} \times \left(\frac{\pi\n,text{rad}}{180^\circ}\right)

    • Radians to degrees: \theta_{\n,text{deg}} = \theta_{\n,text{rad}} \times \left(\frac{180^\circ}{\pi\n,text{rad}}\right)

    • Periodicity: Angles repeat every 360∘360^\circ (2\pi\n,text{rad}):     \theta_{\n,text{equivalent}} = \theta + 360^\circ

  • Axis Identification Case Study:

    • A vector lying directly on the negative y-axis has a direction angle of θ=270∘\theta = 270^\circ counterclockwise relative to the positive x-axis reference line.

Unit Vectors and Right-Handed Coordinate Systems

  • Unit Vector Definition:

    • A unit vector is a dimensionless vector with a magnitude of exactly one (11).

    • Symbolized using a hat accent (i^,j^,k^\hat{i}, \hat{j}, \hat{k}) rather than an arrow.

    • Serves exclusively to specify direction along coordinate axes.

  • Cartesian Unit Vectors:

    • i^\hat{i}: Unit vector pointing along the positive x-axis.

    • j^\hat{j}: Unit vector pointing along the positive y-axis.

    • k^\hat{k}: Unit vector pointing along the positive z-axis.

    • Magnitudes: ∣i^∣=∣j^∣=∣k^∣=1|\hat{i}| = |\hat{j}| = |\hat{k}| = 1.

    • Mutually perpendicular / orthogonal: The angle between any pair of unit vectors (i^\hat{i} and j^\hat{j}, j^\hat{j} and k^\hat{k}, i^\hat{i} and k^\hat{k}) is 90∘90^\circ.

    • Mathematical definition of a component vector along an axis:     x⃗=xi^  ⟹  i^=x⃗∣x⃗∣\vec{x} = x\hat{i} \implies \hat{i} = \frac{\vec{x}}{|\vec{x}|}

  • Right-Handed Coordinate System Convention:

    • Standard coordinate systems in physics and engineering adhere to the Right-Hand Rule.

    • Right-Hand Rule Procedure:

    • Extend the fingers of the right hand along the positive x-axis (i^\hat{i}).

    • Curl the fingers toward the positive y-axis (j^\hat{j}).

    • The extended right thumb defines the positive z-axis direction (k^\hat{k}).

    • Chirality and Parity:

    • Reflecting a right-handed coordinate system in a mirror yields a left-handed coordinate system.

    • Chirality/parity is crucial in physical and biological sciences (e.g., enantiomeric drug molecules binding to target biological receptors).

Cartesian Components and Vector Projections

  • Cartesian Form Representation:

    • Expresses a vector as an algebraic sum of scalar components multiplied by unit vectors:     a⃗=axi^+ayj^\vec{a} = a_x \hat{i} + a_y \hat{j} (in 2D) or a⃗=axi^+ayj^+azk^\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k} (in 3D).

    • axa_x: Scalar component along the x-axis.

    • aya_y: Scalar component along the y-axis.

    • aza_z: Scalar component along the z-axis.

    • Scalar component coefficients ax,ay,aza_x, a_y, a_z can be positive, negative, or zero.

  • Physical Concept of Projections:

    • Component axa_x represents the horizontal projection (shadow) cast by vector a⃗\vec{a} onto the x-axis from vertical overhead sunlight.

    • Component aya_y represents the vertical projection (shadow) cast by vector a⃗\vec{a} onto a vertical screen (y-axis) from horizontal light.

    • The components axa_x and aya_y form the legs of a right-angled triangle whose hypotenuse is the magnitude ∣a⃗∣|\vec{a}|.

Converting Between Polar and Cartesian Forms

  • Polar to Cartesian Conversion:

    • Given magnitude ∣a⃗∣|\vec{a}| and direction angle θ\theta measured counterclockwise from the positive x-axis:     ax=∣a⃗∣cos⁡(θ)a_x = |\vec{a}| \cos(\theta)     ay=∣a⃗∣sin⁡(θ)a_y = |\vec{a}| \sin(\theta)

    • Trigonometric Derivation (SOH CAH TOA):

    • \sin(\theta) = \frac{\n,text{opposite}}{\n,text{hypotenuse}} = \frac{a_y}{|\vec{a}|} \implies a_y = |\vec{a}| \sin(\theta)

    • \cos(\theta) = \frac{\n,text{adjacent}}{\n,text{hypotenuse}} = \frac{a_x}{|\vec{a}|} \implies a_x = |\vec{a}| \cos(\theta)

    • \tan(\theta) = \frac{\n,text{opposite}}{\n,text{adjacent}} = \frac{a_y}{a_x}

    • When θ\theta is measured counterclockwise from the positive x-axis, these equations automatically calculate the correct mathematical signs for axa_x and aya_y across all four quadrants.

  • Polar to Cartesian Practice Example:

    • Given vector a⃗\vec{a} with magnitude 6.0\n,text{units} at θ=120∘\theta = 120^\circ (Quadrant II):

    • a_x = 6.0 \cos(120^\circ) = 6.0 \times (-0.5) = -3.0\n,text{units}

    • a_y = 6.0 \sin(120^\circ) = 6.0 \times \left(\frac{\sqrt{3}}{2}\right) \approx 5.196\n,text{units}

    • Cartesian notation: a⃗=−3.0i^+5.2j^\vec{a} = -3.0\hat{i} + 5.2\hat{j}.

  • Cartesian to Polar Conversion:

    • Given scalar components axa_x and aya_y:

    • Magnitude (Pythagorean Theorem):     ∣a⃗∣=ax2+ay2|\vec{a}| = \sqrt{a_x^2 + a_y^2}

    • Angle Calculation and Quadrant Adjustments:

    • The raw inverse tangent calculation \theta_{\n,text{calc}} = \arctan\left(\frac{a_y}{a_x}\right) returns principal values bounded in (−90∘,90∘)\left(-90^\circ, 90^\circ\right).

    • Quadrant I (a_x > 0, a_y > 0):       θ=arctan⁡(ayax)\theta = \arctan\left(\frac{a_y}{a_x}\right)

    • Quadrant II (a_x < 0, a_y > 0):       θ=arctan⁡(ayax)+180∘\theta = \arctan\left(\frac{a_y}{a_x}\right) + 180^\circ

    • Quadrant III (a_x < 0, a_y < 0):       θ=arctan⁡(ayax)+180∘\theta = \arctan\left(\frac{a_y}{a_x}\right) + 180^\circ

    • Quadrant IV (a_x > 0, a_y < 0):       θ=arctan⁡(ayax)+360∘\theta = \arctan\left(\frac{a_y}{a_x}\right) + 360^\circ