2.2 Vectors in Three Dimensions

Three-dimensional coordinate systems expand our understanding of space by adding a z-axis to the familiar x and y axes. This allows us to represent points with three coordinates (x, y, z) and introduces new concepts like octants and 3D distance formulas.

Vector operations in three dimensions build on 2D concepts, introducing addition, subtraction, and scalar multiplication in 3D space. These operations have geometric interpretations, helping us visualize how vectors interact and change in three-dimensional environments.

Three-Dimensional Coordinate Systems and Vectors

Three-dimensional coordinate systems

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• Extends the two-dimensional coordinate system by adding a third perpendicular axis (z-axis)
  • x-axis, y-axis, and z-axis intersect at the origin (0, 0, 0)
  • Each point in 3D space represented by an ordered triple (x, y, z) with coordinates along respective axes
• Three coordinate planes formed by the intersection of the axes
  • xy-plane contains the x-axis and y-axis
  • xz-plane contains the x-axis and z-axis
  • yz-plane contains the y-axis and z-axis
• Three-dimensional space divided into eight octants by the coordinate planes numbered from I to VIII (similar to quadrants in 2D space)

Distances in 3D space

• Distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ given by the three-dimensional distance formula:
  • $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
  • Extends the Pythagorean theorem to three dimensions
• Distance between a point and a plane is the perpendicular distance from the point to the plane
  • Find the projection of the vector from the point to any point on the plane onto the normal vector of the plane

Equations of planes and spheres

• General equation of a plane in 3D space:
  • $Ax + By + Cz + D = 0$, where $A$, $B$, $C$, and $D$ are constants, and $(A, B, C) \neq (0, 0, 0)$
  • Vector $\vec{n} = (A, B, C)$ is the normal vector of the plane, perpendicular to any vector lying on the plane
• Equation of a plane passing through a point $P_0(x_0, y_0, z_0)$ with a normal vector $\vec{n} = (A, B, C)$:
  • $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$
• Equation of a sphere with center $C(x_0, y_0, z_0)$ and radius $r$:
  • $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2$
  • Represents the set of all points in 3D space at a distance $r$ from the center $C$

Vector Operations in Three Dimensions

Vector operations in ℝ³

• Vectors in 3D space represented by ordered triples $\vec{u} = (u_1, u_2, u_3)$ and $\vec{v} = (v_1, v_2, v_3)$
• Vector addition: $\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3)$
  • Geometrically follows the parallelogram law (similar to vector addition in 2D space)
• Vector subtraction: $\vec{u} - \vec{v} = (u_1 - v_1, u_2 - v_2, u_3 - v_3)$
  • Subtracting $\vec{v}$ from $\vec{u}$ is equivalent to adding the negative of $\vec{v}$ to $\vec{u}$
• Scalar multiplication: For a scalar $c$ and a vector $\vec{u} = (u_1, u_2, u_3)$, $c\vec{u} = (cu_1, cu_2, cu_3)$
  • Changes the magnitude of the vector while maintaining its direction if $c > 0$, reverses direction if $c < 0$

Geometry of vector operations

• Vector addition in 3D space:
  • Place the initial point of the second vector at the terminal point of the first vector
  • Draw the resultant vector from the initial point of the first vector to the terminal point of the second vector
• Vector subtraction in 3D space:
  • Place the initial points of both vectors at the same point
  • Draw the resultant vector from the terminal point of the second vector to the terminal point of the first vector
• Scalar multiplication in 3D space:
  • Scales the vector by the factor $c$, changing its magnitude but not its direction (if $c > 0$)
    1. If $c > 1$, the vector is stretched
    2. If $0 < c < 1$, the vector is compressed
    3. If $c = 1$, the vector remains unchanged
    4. If $c = 0$, the vector becomes the zero vector
    5. If $c < 0$, the vector is scaled and its direction is reversed

Vector components and representations

• Vector components: The scalar values that represent a vector along each coordinate axis (x, y, and z components)
• Unit vector: A vector with a magnitude of 1, often used to represent direction
• Vector projection: The projection of one vector onto another, useful in decomposing vectors
• Linear combination: Expressing a vector as a sum of scalar multiples of other vectors, fundamental in understanding vector spaces