11.2 Vectors in Space

Core concept: vectors in space

• A vector is a quantity with both magnitude and direction.
• Example: a velocity vector of an object at time t points in the direction of travel.
• The length (magnitude) of the velocity vector gives the object's speed at time t.
• Vectors are represented visually as directed line segments from a tail point to a head (terminal point).

Points, tails, heads, and notation

• A point in space is written as P(x, y, z).
• In general, two points define a vector: a tail (initial point) and a head (terminal point).
• If the tail is at P0(x0, y0, z0) and the head is at P1(x1, y1, z1), the vector from P0 to P1 is denoted as
  $\vec{P<em>0P</em>1} = \langle x<em>1 - x</em>0,\; y<em>1 - y</em>0,\; z<em>1 - z</em>0 \rangle.$
• The vector can also be referred to using angle brackets ⟨ , ⟩ or with bold/arrow notation: $\vec{v},\; \mathbf{v}$.
• The term “terminal point” (head) and “initial point” (tail) specify the direction of the vector.
• A position vector of a point P(x, y, z) from the origin is written as
  $\vec{r} = \langle x, y, z \rangle.$

Displacement vs. position

• Displacement from P0 to P1 is given by the vector
  $\vec{d} = \langle x<em>1 - x</em>0,\; y<em>1 - y</em>0,\; z<em>1 - z</em>0 \rangle = \vec{P<em>0P</em>1}.$
• The displacement represents the change in position; the position vector represents the location of a point relative to the origin.

Magnitude and speed

• For a vector $\vec{v} = \langle v<em>x, v</em>y, v<em>z \rangle$, its magnitude (norm) is $|\vec{v}| = \sqrt{v</em>x^2 + v<em>y^2 + v</em>z^2}.$
• If $\vec{v}$ is a velocity vector, then the speed is given by the magnitude:
  $\text{speed} = |\vec{v}|.$

Examples and computation

• Example 1: Let P0 = (2, -1, 4) and P1 = (5, 3, 1). Then
  $\vec{P<em>0P</em>1} = \langle 5-2,\; 3-(-1),\; 1-4 \rangle = \langle 3,\; 4,\; -3 \rangle.$
  Magnitude: $|\vec{P<em>0P</em>1}| = \sqrt{3^2 + 4^2 + (-3)^2} = \sqrt{9 + 16 + 9} = \sqrt{34}.$
• Example 2: The position vector of a point P = (x, y, z) is $\vec{r} = \langle x, y, z \rangle.$

Visual interpretation

• A vector is represented as a directed line segment oriented from the tail to the head.
• The length of the segment corresponds to the magnitude; the direction from tail to head indicates the vector's direction.
• The components along the x, y, and z axes are the changes (\Delta x, \Delta y, \Delta z) between the two points.

Notation and special terminology

• Point vs. vector: a point like P0 or P1 denotes a location; a vector is the directed difference between two points.
• The bracket notation (e.g., ⟨a, b, c⟩) indicates a vector; the same object can be written as \vec{v} or \mathbf{v} depending on the convention.

Connections to broader concepts

• Vectors generalize from 2D to 3D: the same definitions apply with three coordinates (x, y, z).
• In physics and engineering, vectors model quantities with both magnitude and direction (velocity, force, acceleration).
• Key distinction: position vectors (locations) vs. displacement vectors (changes in position).

Practical implications

• When combining motions or forces, vector addition uses component-wise addition: for vectors \vec{a} = ⟨ax, ay, az⟩ and \vec{b} = ⟨bx, by, bz⟩, their sum is
  $\vec{a} + \vec{b} = \langle a<em>x + b</em>x,\; a<em>y + b</em>y,\; a<em>z + b</em>z \rangle.$
• Magnitude calculations are essential for determining speeds and other scalar quantities derived from vectors.