Vector Fundamentals: Components, Magnitude, Direction, and Dot Product

Key concepts: Vector representation vs coordinate systems

• A vector is a quantity with both magnitude and direction; it is independent of the coordinate system. Its components, however, depend on how you orient the axes.
• Graphical view: A vector can be drawn in the plane with horizontal (x) and vertical (y) components, forming a right triangle with the vector as the hypotenuse.
• Components tell you how much of the vector lies along each axis; their signs depend on the chosen axes.
• If you slide a vector without rotating it, its direction and magnitude stay the same even though its tail and head positions may change; this is translation invariance of vectors.

From components to magnitude and direction

• Suppose a vector a has components
  • $a<em>x$ along the x-axis and $a</em>y$ along the y-axis.
• Magnitude (length) of the vector:
  • $|\mathbf{a}| = \sqrt{a<em>x^2 + a</em>y^2}$
• Direction (angle with respect to the +x axis):
  • Basic relation: $\tan \theta = \frac{a<em>y}{a</em>x}$
  • Therefore, using arctangent:
  • $\theta = \arctan\left(\frac{a<em>y}{a</em>x}\right)$
  • Note: you must account for the quadrant of (ax, ay). If only using the basic arctan, you may need to add/subtract 180° (or use the two-argument arctangent function) to place the angle in the correct quadrant.
  • A robust way: $\theta = \operatorname{atan2}(a<em>y, a</em>x)$ which gives the correct angle in \,[0, 360°) or \,[-\pi, \pi).
• Summary: given components, you can get magnitude and direction; given magnitude and direction, you can get components via
  • $a<em>x = |\mathbf{a}| \cos \theta, \quad a</em>y = |\mathbf{a}| \sin \theta.$

From magnitude and direction to components

• If you know the magnitude and the direction of a, use:
  • $a_x = |\mathbf{a}| \cos \theta$
  • $a_y = |\mathbf{a}| \sin \theta$
• Example: If $|\mathbf{a}| = 5$ and $\theta = 53.13°$, then
  • $a_x = 5 \cos(53.13°) \approx 3$
  • $a_y = 5 \sin(53.13°) \approx 4$

Quadrants and angle adjustments (practical tip)

• If both components are positive, the vector lies in the first quadrant; the computed angle should be between 0° and 90°.
• If the arctangent result is not in the correct quadrant, adjust by ±180° to place the angle properly.
• A reliable practice: use $\theta = \operatorname{atan2}(a<em>y, a</em>x)$ to avoid manual quadrant adjustments.

Coordinate systems and the same vector

• The vector itself is invariant under a change of coordinate system.
• Its components change when you rotate or shift the axes.
• Different coordinate systems can make some operations easier:
  • For example, certain additions or multiplications of vectors may be easier in one system due to simpler components.
• Practical takeaway: pick a coordinate system that simplifies your calculations, but remember the underlying vector is the same.

Scalar multiplication (scaling a vector)

• If you scale a vector by a scalar c:
  • $\mathbf{d} = c\,\mathbf{a}$
  • Components scale individually:
  • $d<em>x = c \; a</em>x, \ d<em>y = c \; a</em>y$
  • Magnitude scales by |c|: $|\mathbf{d}| = |c| \; |\mathbf{a}|$
  • Direction stays the same if c > 0; reverses if c < 0 (points opposite).
• Geometric interpretation: this follows from similar triangles when you scale the vector's length without changing its direction.
• Example: If $\mathbf{a} = (a<em>x, a</em>y) = (3,4)$ and $c = 2\,,$ then
  • $\mathbf{d} = (6, 8)$
  • $|\mathbf{d}| = 2|\mathbf{a}|$

Vector addition and subtraction (component form)

• Given two vectors with components $\mathbf{a} = (a<em>x, a</em>y)$ and $\mathbf{b} = (b<em>x, b</em>y)$:
  • Sum: $\mathbf{a} + \mathbf{b} = (a<em>x + b</em>x, \; a<em>y + b</em>y)$
  • Subtract: $\mathbf{a} - \mathbf{b} = (a<em>x - b</em>x, \; a<em>y - b</em>y)$
• Graphical intuition: add the x-components along the x-axis and the y-components along the y-axis. The result is the vector from the tail of a to the head of the combined tail-to-head construction.
• Examples from typical quizzes (methods): reading x-components from diagrams by projecting onto the x-axis; translating vectors to start at the origin can simplify reading components, but the vector properties do not depend on the starting point.
• Important note: origin location does not affect the vector’s properties; only the components change with the coordinate orientation.

Reading components from diagrams (practice insight)

• To determine an x-component from a diagram: measure the horizontal displacement from the tail to the head along the x-axis direction.
• The same vector might have different component values in a differently oriented coordinate system, even though the vector itself is unchanged.
• Subtlety: when combining vectors graphically, you can translate them as needed to apply the tail-to-head method without changing the result.

Dot product (scalar product): definition and interpretation

• There are two standard vector products in 3D, but this course focuses on the scalar (dot) product:
  • Definition (geometric): for vectors $\mathbf{a}, \mathbf{b}$, the scalar product is
  • $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \; |\mathbf{b}| \; \cos \theta$
    where $\theta$ is the angle between the vectors.
  • In terms of components (2D):
  • $\mathbf{a} \cdot \mathbf{b} = a<em>x b</em>x + a<em>y b</em>y$
• Geometric derivation (outline): decompose $\mathbf{a}$ into a parallel component to $\mathbf{b}$ and a perpendicular component to $\mathbf{b}$.
  • The parallel component has length $|\mathbf{a}| \cos \theta$ along the direction of $\mathbf{b}$.
  • The dot product with $\mathbf{b}$ equals the projection length of $\mathbf{a}$ onto $\mathbf{b}$ times the magnitude of $\mathbf{b}$:
  • $\mathbf{a} \cdot \mathbf{b} = (|\mathbf{a}| \cos \theta) \; |\mathbf{b}| = |\mathbf{a}| \; |\mathbf{b}| \cos \theta.$
• Significance and uses: the dot product measures how much one vector extends in the direction of another; it is zero if vectors are orthogonal, positive if acute, negative if obtuse.
• Note on the cross product: there exists another vector product (the cross product) that yields a vector, but it is not covered in this course.

Quick practice problems and verification ideas

• Example 1: Given $\mathbf{a} = (3, 4)$.
  • Magnitude: $|\mathbf{a}| = \sqrt{3^2 + 4^2} = 5$
  • Direction: $\theta = \operatorname{atan2}(4, 3) \approx 53.13°$
  • Alternative: $\theta = \arctan\left(\frac{4}{3}\right) \approx 0.93\text{ rad}$ (then adjust for quadrant as needed).
• Example 2: Scalar multiplication
  • If $\mathbf{d} = 2\,\mathbf{a}$ with $\mathbf{a} = (3,4)$, then $\mathbf{d} = (6,8)$ and $|\mathbf{d}| = 2|\mathbf{a}| = 10$.
• Example 3: Addition
  • If $\mathbf{a} = (3,4)$ and $\mathbf{b} = (1,-2)$, then $\mathbf{a} + \mathbf{b} = (4,2)$.
• Example 4: Dot product
  • If $\mathbf{a} = (2,3)$ and $\mathbf{b} = (4,0)$, then $\mathbf{a} \cdot \mathbf{b} = 2\cdot 4 + 3\cdot 0 = 8.$
• Example 5: Magnitude-direction to components
  • If $|\mathbf{a}| = 5$ and $\theta = 45°$, then $a<em>x = 5\cos 45° = \frac{5}{\sqrt{2}} \approx 3.54$, $a</em>y = 5\sin 45° = \frac{5}{\sqrt{2}} \approx 3.54$.

Key formulas at a glance

• Magnitude from components: $|\mathbf{a}| = \sqrt{a<em>x^2 + a</em>y^2}$
• Direction from components: $\theta = \operatorname{atan2}(a<em>y, a</em>x)$
• Components from magnitude/direction: $a<em>x = |\mathbf{a}| \cos \theta, \quad a</em>y = |\mathbf{a}| \sin \theta$
• Dot product (components): $\mathbf{a} \cdot \mathbf{b} = a<em>x b</em>x + a<em>y b</em>y$
• Dot product (geometric): $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \; |\mathbf{b}| \cos \theta$
• Scalar multiplication: $\mathbf{d} = c\mathbf{a} \Rightarrow d<em>x = c a</em>x, \; d<em>y = c a</em>y$
• Vector addition: $\mathbf{a} + \mathbf{b} = (a<em>x + b</em>x, \; a<em>y + b</em>y)$
• Vector subtraction: $\mathbf{a} - \mathbf{b} = (a<em>x - b</em>x, \; a<em>y - b</em>y)$