Home / Courses / Calculus III (Multivariable) / Unit 1

Calculus III (Multivariable) Unit 1: Vectors & the geometry of space

Multivariable calculus starts in 3D space. Vectors describe direction and size, and dot and cross products measure angles, areas and perpendicular directions.

Covers: 3D coordinates, dot & cross product, lines & planes, quadric surfaces

Study guide

Multivariable calculus works in three dimensions. Points are (x, y, z), and the distance between two points is √(Δx² + Δy² + Δz²). Vectors ⟨a, b, c⟩ have magnitude √(a² + b² + c²) and can be added, scaled and normalized into unit vectors.

The dot product u·v = u₁v₁ + u₂v₂ + u₃v₃ = |u||v|cos θ is a number. It finds angles between vectors, and u·v = 0 means they're perpendicular. It also gives projections: the component of one vector along another.

The cross product u × v is a vector perpendicular to both u and v, found with a 3×3 determinant. Its direction follows the right-hand rule, and its magnitude |u × v| = |u||v|sin θ equals the area of the parallelogram the vectors span. It's not commutative: v × u = −(u × v).

A line in space through r₀ with direction v is r(t) = r₀ + tv, or in parametric form x = x₀ + at, etc. A plane through (x₀, y₀, z₀) with normal vector ⟨a, b, c⟩ has equation a(x − x₀) + b(y − y₀) + c(z − z₀) = 0. Cross two vectors in a plane to get its normal.

Quadric surfaces are 3D graphs of second-degree equations: spheres, ellipsoids, paraboloids, cones, cylinders and hyperboloids. Identify them by their traces, the curves formed by slicing with planes x = k, y = k or z = k.

The distance from a point P to the plane through Q with normal n is |PQ·n|/|n|. For the plane ax + by + cz = d and the point (x₁, y₁, z₁), that is |ax₁ + by₁ + cz₁ − d|/√(a² + b² + c²). To find where a line meets a plane, substitute the line's parametric equations into the plane's equation and solve for t. A line is parallel to a plane when its direction vector is perpendicular to the plane's normal (dot product 0), and perpendicular to the plane when the direction is a multiple of the normal.

A 2×2 determinant is ad − bc, and a 3×3 determinant expands along a row into three 2×2 determinants. A square matrix with determinant 0 is singular: its rows lie in a common plane, and Ax = 0 has nonzero solutions, such as the cross product of two rows of a 3×3 singular matrix. To multiply a matrix A by a vector x, take the dot product of each row of A with x. A is invertible exactly when det A ≠ 0, and then the system Ax = b has the single solution x = A⁻¹b.

The area of a triangle is half the magnitude of the cross product of two edge vectors. The absolute value |u·(v × w)| equals the volume of the parallelepiped the three vectors span, and equals the 3×3 determinant with rows u, v, w.

Key ideas

• Points in 3D are (x, y, z); distance generalizes the Pythagorean theorem.
• Dot product gives a number and measures how aligned two vectors are; zero means perpendicular.
• Cross product gives a vector perpendicular to both, with length equal to the parallelogram's area.
• A line needs a point and a direction vector; a plane needs a point and a normal vector.

Formulas and rules

Common mistakes

• Mixing up which product gives a scalar and which gives a vector.
• Using a direction vector as a plane's normal.

Words to know

Orthogonal vectors

u · v = 0: the vectors are perpendicular.

Cross product

u × v: a vector perpendicular to both u and v.

Magnitude of the cross product

|u × v| = area of the parallelogram spanned by u and v.

Equation of a plane through (x₀, y₀, z₀) with normal ⟨a, b, c⟩

a(x − x₀) + b(y − y₀) + c(z − z₀) = 0

Vector equation of a line

r(t) = r₀ + t·v

Distance in 3D

√(Δx² + Δy² + Δz²)

Unit vector

A vector of length 1: v/|v|.

Dot product

u·v = u₁v₁ + u₂v₂ + u₃v₃ = |u||v|cos θ

Scalar projection

comp_u v = (u·v)/|u|

Right-hand rule

Gives the direction of u × v.

Normal vector (plane)

A vector perpendicular to the plane, ⟨a, b, c⟩ in ax + by + cz = d.

Quadric surface

The 3D graph of a second-degree equation, like an ellipsoid or paraboloid.

Trace

The curve where a surface meets a plane such as z = k.

3D coordinate system

Points (x, y, z) on three perpendicular axes, using the right-hand orientation.

Sample practice questions

41 more questions like these, from level 1 to 5, are in the practice set.