Surfaces, Distance, Spheres & Vectors – Lecture Vocabulary

Vocabulary flashcards covering planes, cylinders, distance, spheres, vectors, dot product, projections, and work as presented in the lecture notes.

Plane x = k (in ℝ³)

A plane parallel to the yz-plane located k units from it.

Plane y = k (in ℝ³)

A plane parallel to the xz-plane located k units from it.

Plane z = k (in ℝ³)

A plane parallel to the xy-plane located k units from it.

yz-plane

The coordinate plane given by x = 0 in ℝ³.

xz-plane

The coordinate plane given by y = 0 in ℝ³.

xy-plane

The coordinate plane given by z = 0 in ℝ³.

Vertical plane 3x − y = −3

A plane perpendicular to the xy-plane that intersects it along the line 3x − y = −3.

Cylinder x² + y² = 4

A circular cylinder of radius 2 centered on the z-axis.

Cylinder y² + z² = 4

A circular cylinder of radius 2 centered on the x-axis.

Distance Formula (ℝ³)

Distance between P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂) is √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²].

Sphere (standard equation)

(x−h)² + (y−k)² + (z−l)² = r² represents a sphere with center (h,k,l) and radius r.

Vector

A quantity with both magnitude and direction, often pictured as an arrow.

Scalar

A quantity having magnitude only, no direction (e.g., mass, temperature).

Zero vector

The unique vector of length 0 with no specific direction.

Vector magnitude (length)

For a = ⟨a₁,a₂,a₃⟩, |a| = √(a₁² + a₂² + a₃²).

Unit vector

A vector whose magnitude is 1; û = a / |a| gives the unit vector in direction of a.

Standard basis vectors (ℝ³)

î = ⟨1,0,0⟩, ĵ = ⟨0,1,0⟩, k̂ = ⟨0,0,1⟩.

Vector addition

Sum of vectors found by adding corresponding components: ⟨a₁,b₁⟩+⟨a₂,b₂⟩=⟨a₁+a₂,b₁+b₂⟩.

Scalar multiplication

Multiplying vector a by scalar c scales magnitude by |c| and may reverse direction if c<0.

Dot product (ℝ³)

For a•b = a₁b₁ + a₂b₂ + a₃b₃; result is a scalar.

Angle between vectors

cos θ = (a•b) / (|a||b|) for non-zero vectors a and b.

Parallel vectors

Vectors satisfying b = c a for some non-zero scalar c (angle 0 or π).

Orthogonal vectors

Vectors whose dot product is zero (angle 90°).

Scalar projection (compₐb)

Signed magnitude of b in direction of a: (a•b)/|a|.

Vector projection (projₐb)

Component vector of b along a: [(a•b)/|a|²] a.

Work (vector form)

W = F • d = |F||d|cos θ, work done by constant force F moving object displacement d.

Triangle Law of Addition

Placing the tail of v at the head of u makes u+v the vector from u's tail to v's head.

Parallelogram Law

u+v is the diagonal of the parallelogram formed by u and v with a common tail.

Position vector

Vector whose initial point is the origin and terminal point is (a₁,a₂,a₃); represents that point.

Vector components

The individual scalar entries (a₁,a₂,a₃) that describe a vector in coordinate form.