Key Concepts of Vectors in 3D Space

Vectors in 3D space are essential for understanding motion and geometry. They represent quantities with both direction and magnitude, allowing us to perform operations like addition and scalar multiplication, which are crucial in Calculus II for analyzing curves and surfaces.

1. Definition of vectors in 3D space

   • A vector in 3D space is represented as an ordered triplet (x, y, z), indicating its position relative to the origin.
   • Vectors have both magnitude (length) and direction, distinguishing them from scalars, which only have magnitude.
   • Vectors can be visualized as arrows in a three-dimensional coordinate system.

2. Vector operations (addition, subtraction, scalar multiplication)

   • Addition: Vectors are added component-wise; for vectors A = (a1, a2, a3) and B = (b1, b2, b3), A + B = (a1 + b1, a2 + b2, a3 + b3).
   • Subtraction: Similar to addition, vectors are subtracted component-wise; A - B = (a1 - b1, a2 - b2, a3 - b3).
   • Scalar multiplication: A vector can be multiplied by a scalar k, resulting in kA = (ka1, ka2, ka3), scaling its magnitude without changing its direction.

3. Dot product and its properties

   • The dot product of two vectors A and B is calculated as A · B = a1b1 + a2b2 + a3b3, yielding a scalar.
   • It measures the cosine of the angle between the two vectors, indicating how aligned they are.
   • Properties include commutativity (A · B = B · A) and distributivity (A · (B + C) = A · B + A · C).

4. Cross product and its properties

   • The cross product of two vectors A and B results in a new vector C = A × B, which is perpendicular to both A and B.
   • The magnitude of the cross product is given by |C| = |A||B|sin(θ), where θ is the angle between A and B.
   • Properties include anti-commutativity (A × B = -B × A) and the right-hand rule for determining the direction of the resulting vector.

5. Vector equations of lines

   • A line in 3D can be represented by the vector equation r(t) = r0 + tv, where r0 is a point on the line and v is the direction vector.
   • The parameter t varies over all real numbers, allowing the equation to describe all points on the line.
   • This representation is useful for visualizing and calculating intersections with other geometric objects.

6. Vector equations of planes

   • A plane can be defined by a point r0 on the plane and a normal vector n, expressed as r · n = r0 · n.
   • Alternatively, a plane can be represented using two direction vectors, r(t, s) = r0 + tA + sB, where A and B are non-parallel vectors in the plane.
   • This formulation allows for easy manipulation and intersection calculations with lines and other planes.

7. Parametric equations of lines and planes

   • Parametric equations express the coordinates of points on a line or plane as functions of one or more parameters.
   • For a line, x = x0 + at, y = y0 + bt, z = z0 + ct, where (x0, y0, z0) is a point on the line and (a, b, c) is the direction vector.
   • For a plane, x = x0 + sA1 + tA2, y = y0 + sB1 + tB2, z = z0 + sC1 + tC2, where A and B are direction vectors in the plane.

8. Distance between points, lines, and planes

   • The distance between two points in 3D is calculated using the distance formula: d = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²).
   • The distance from a point to a line can be found using the projection of the vector from the point to a point on the line.
   • The distance from a point to a plane is given by the formula: d = |Ax0 + By0 + Cz0 + D| / √(A² + B² + C²), where Ax + By + Cz + D = 0 is the plane equation.

9. Projections of vectors

   • The projection of vector A onto vector B is given by proj_B(A) = (A · B / |B|²)B, resulting in a vector parallel to B.
   • This operation helps in decomposing vectors into components along specified directions.
   • The concept of projection is crucial in applications such as physics, where forces are resolved into components.

10. Vector-valued functions and space curves

    • A vector-valued function assigns a vector to each point in its domain, typically expressed as r(t) = (x(t), y(t), z(t)).
    • Space curves are traced by the path of a vector-valued function as the parameter t varies, allowing for the study of motion in three dimensions.
    • Derivatives of vector-valued functions provide information about the velocity and acceleration of the curve, essential in physics and engineering applications.