← Calculus III (Multivariable)

Unit 1: Vectors and the Geometry of Space

Points and vectors, and their two products, build the lines, planes, quadric surfaces, and curved coordinate grids every later chapter sits inside.

23 topics · 89 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

2.1 The Three-Dimensional Coordinate System

  1. How it worksHow do the x, y, and z axes split space into eight separate octants?
  2. Worked exampleFind the distance between the points (1, -2, 4) and (3, 5, -1).
  3. Worked exampleFind the equation of the sphere centered at (2, -1, 3) with radius 5.
  4. Common mistakeWhy isn't x2 + y2 = 9 a circle once you're graphing it in three dimensions?
Your turn1 of 2
Find the distance between the points (2, 2, 5) and (-3, -5, -4).

2.1 Vectors and Vector Algebra

  1. How it worksHow do you add two vectors by stacking their matching components?
  2. How it worksHow does scalar multiplication stretch or reverse a vector without ever changing its line?
  3. Worked exampleFind 3a - 2b for a = (2, -1) and b = (4, 3).
  4. Common mistakeWhy isn't a vector the same thing as the point sitting at its tip?
Your turn1 of 2
For a = (3, 2) and b = (7, -4), find the x-component of 3a - 2b.

2.2 Vectors in Three Dimensions and the Standard Basis

  1. How it worksHow do the unit vectors i, j, and k combine to write any vector in component form?
  2. Worked exampleFind the vector from P(1, 0, 2) to Q(4, -3, 5) written in terms of i, j, and k.
  3. Going furtherWhat happens to the i, j, k decomposition once you move to a vector in four dimensions?
  4. Ties togetherHow does writing a vector in i, j, k form connect to writing a point's own position vector from the origin?
Your turn1 of 2
Find the k-component (z-component) of the vector from P(2, 2, 5) to Q(-3, -5, -4).

2.2 Magnitude and Unit Vectors

  1. Worked exampleFind the magnitude of v = (3, -4, 12).
  2. How it worksHow do you shrink any nonzero vector down to a unit vector that still points the same way?
  3. Worked exampleFind a unit vector in the same direction as w = (1, 2, -2).
  4. Common mistakeWhy doesn't dividing a vector's components by its own length ever change the direction it points?
Your turn1 of 2
Find the magnitude of v = (4, 2, 7).

2.2 Direction Angles and Direction Cosines

  1. How it worksHow do the direction cosines of a vector come directly from its unit vector's components?
  2. Worked exampleFind the direction angles of the vector v = (1, 2, 2).
  3. Common mistakeWhy don't a vector's three direction angles add up to 90 or 180 degrees the way triangle angles do?
  4. Going furtherWhat happens to a vector's direction cosines when the vector is scaled by a positive constant?
Your turn1 of 2
Find the direction angle, in degrees, that v = (4, 1, 4) makes with the positive x-axis.

2.3 The Dot Product and the Angle Between Vectors

  1. How it worksHow does the dot product formula hide the cosine of the angle between two vectors inside it?
  2. Worked exampleFind the angle between u = (1, 1, 0) and v = (1, 0, 1).
  3. Common mistakeWhy isn't the dot product of two vectors itself a vector?
  4. Going furtherWhat happens to the dot product of two fixed-length vectors as the angle between them grows past 90 degrees?
Your turn1 of 2
Find the angle, in degrees, between u = (4, 1, 4) and v = (2, -4, -3).

2.3 Testing Vectors for Orthogonality

  1. Worked exampleShow that u = (2, -3, 1) and v = (1, 1, 1) are orthogonal.
  2. Worked exampleFind the value of k that makes (k, 2, -1) orthogonal to (3, -1, 4).
  3. Going furtherWhat happens to the orthogonality test the moment one of the two vectors is the zero vector?
  4. Ties togetherHow does the cross product give a quick way to build a vector orthogonal to two given vectors at once?
Your turn1 of 2
Find the value of k that makes (k, 2, 2) orthogonal to (6, -3, -5).

2.3 Vector Projections and Work

  1. How it worksHow does projecting one vector onto another split it into a parallel piece and a perpendicular piece?
  2. Worked exampleFind the projection of a = (3, 4) onto b = (1, 0).
  3. Worked exampleFind the work done by a 50 N force in the direction (3, 4) moving an object along the displacement (6, 0) meters.
  4. Common mistakeWhy isn't the work done by a force just its magnitude times the distance the object travels?
Your turn1 of 2
Find the x-component of the vector projection of a = (3, 2) onto b = (6, -3).

2.4 The Cross Product and Its Geometric Meaning

  1. How it worksHow does the right-hand rule tell you which way the cross product vector ends up pointing?
  2. Worked exampleFind u x v for u = (1, 0, 0) and v = (0, 1, 0).
  3. Common mistakeWhy isn't the cross product commutative the way ordinary multiplication is?
  4. Going furtherWhat happens to the cross product of two vectors as the angle between them shrinks toward 0?
Your turn1 of 2
Find the i-component (x-component) of u x v for u = (2, 2, 5) and v = (-3, -5, -4).

2.4 Area of a Parallelogram and Torque

  1. How it worksHow does the magnitude of the cross product equal the area of the parallelogram two vectors span?
  2. Worked exampleFind the area of the parallelogram spanned by a = (2, 1, 0) and b = (0, 3, 1).
  3. Worked exampleFind the torque produced by a 40 N force applied at the position vector (0, 2, 0) meters from a bolt.
  4. Going furtherWhat happens to the torque when the same force is applied straight down the length of the handle instead?
Your turn1 of 2
Find the area of the parallelogram spanned by a = (2, 2, 5) and b = (-3, -5, -4).

2.4 The Scalar Triple Product and Volume of a Parallelepiped

  1. How it worksHow does stacking a dot product on top of a cross product end up measuring a three-dimensional volume?
  2. Worked exampleFind the volume of the parallelepiped spanned by (1, 0, 0), (0, 2, 0), and (1, 1, 3).
  3. Common mistakeWhy does a scalar triple product of zero mean the three vectors are flat together, not that one of them is zero?
  4. Going furtherWhat happens to the scalar triple product's value when the three vectors are relabeled in a different order?
Your turn1 of 2
Find the volume of the parallelepiped spanned by a = (2, 1, 4), b = (-2, -4, -3), and c = (2, -5, -3).

2.5 Parametric and Symmetric Equations of a Line in Space

  1. How it worksHow do a point and a direction vector turn into the parametric equations of a line?
  2. Worked exampleFind parametric equations for the line through (1, 2, -1) parallel to (3, 0, 2).
  3. Worked exampleFind the symmetric equations of the line through (0, 1, 4) and (2, -3, 5).
  4. Going furtherWhat happens to a line's parametric equations when its direction vector is scaled by 5?
Your turn1 of 2
A line passes through (2, 2, 5) with direction vector (-3, -5, 2). Write its z-coordinate as a function of the parameter t.

2.5 The Equation of a Plane in Space

  1. How it worksHow does a point together with a normal vector combine into the equation of a plane?
  2. Worked exampleFind the equation of the plane through (1, 0, 2) with normal vector (2, -1, 3).
  3. Worked exampleFind the equation of the plane through the three points (1, 0, 0), (0, 1, 0), and (0, 0, 1).
  4. Common mistakeWhy doesn't a plane's equation change when every one of its coefficients gets multiplied by the same number?
Your turn1 of 2
A plane passes through (2, 2, 5) with normal vector (-3, -5, 2). Solve its equation for z as a function of x and y.

2.5 Intersecting Lines and Planes, and Distance Formulas

  1. Worked exampleFind the point where the line x = 1 + t, y = 2t, z = 3 - t meets the plane x + y + z = 6.
  2. Worked exampleFind the distance from the point (1, 2, 3) to the plane 2x - y + 2z = 4.
  3. Going furtherWhat happens to the line-plane intersection formula once the line runs exactly parallel to the plane?
Your turn1 of 2
Find the distance from the point (2, 2, 5) to the plane 2x + -5y + -4z = 3.

2.5 Finding the Line Where Two Planes Meet

  1. How it worksHow does crossing the two planes' normal vectors hand you the direction of their line of intersection?
  2. Worked exampleFind the line of intersection of the planes x + y + z = 3 and x - y + 2z = 2.
  3. Common mistakeWhy doesn't finding just one point on each plane tell you the line where the two planes intersect?
  4. Going furtherWhat happens to the line-of-intersection formula once the two planes turn out to be parallel?
Your turn1 of 2
Two planes have normal vectors n1 = (2, 2, 5) and n2 = (-3, -5, -4). Find the x-component of the direction vector of their line of intersection, n1 x n2.

2.6 Quadric Surfaces: Ellipsoids and Paraboloids

  1. How it worksHow do the horizontal traces of an elliptic paraboloid stack up into a bowl shape?
  2. Worked exampleSketch and identify the surface x2/4 + y2/9 + z2 = 1.
  3. Common mistakeWhy isn't a paraboloid just a parabola spun once around an axis and left at that?
  4. Worked exampleSketch and identify the surface z = x2 - y2 and describe why it's called a saddle.
Your turn1 of 2
For the elliptic paraboloid z = x2/4 + y2/9, what shape is the horizontal trace (cross-section) at height z = k for k > 0?

2.6 Quadric Surfaces: Hyperboloids and Cones

  1. How it worksHow does flipping one sign in a quadric equation turn an ellipsoid into a hyperboloid of one sheet?
  2. Worked exampleSketch and identify the surface x2 + y2 - z2 = 1 by examining its traces.
  3. Worked exampleSketch and identify the surface z2 = x2 + y2 and describe its cross sections.
  4. Going furtherWhat happens to a hyperboloid of one sheet as the constant on its right-hand side shrinks toward zero?
Your turn1 of 2
The equation x2/42 + y2/42 - z2/52 = 1 describes a hyperboloid of ONE sheet. If the sign on the x2 term is also flipped to negative, giving -x2/42 + y2/42 - z2/52 = 1, what surface results?

2.6 Classifying a Quadric Surface From Its Traces

  1. How it worksHow does counting how many squared terms are positive, negative, or missing sort an equation into a quadric family?
  2. Worked exampleFind the type of surface 4x2 - y2 + z2 = 4 describes and sketch its traces.
  3. Ties togetherHow does this classification method connect back to recognizing conic sections from their equations in the plane?
Your turn1 of 2
The equation 6x2 - y2 + z2 = 6 (with 6 > 0) describes which quadric surface?

2.7 Cylindrical Coordinates

  1. How it worksHow do r, theta, and z convert back and forth into x, y, and z?
  2. Worked exampleFind the cylindrical coordinates of the point (2, 2, 5).
  3. Worked exampleFind the Cartesian coordinates of the cylindrical point (4, π/3, -1).
  4. Common mistakeWhy isn't r in cylindrical coordinates allowed to soak up the sign that theta would otherwise carry?
Your turn1 of 2
Find the cylindrical radius r for the Cartesian point (3, 2, 7).

2.7 Spherical Coordinates

  1. How it worksHow do rho, theta, and phi locate a point by its distance from the origin and two angles?
  2. Worked exampleFind the spherical coordinates of the point (1, 1, 1).
  3. Worked exampleFind the Cartesian coordinates of the spherical point (2, π/4, π/3).
  4. Going furtherWhat happens to a point's spherical coordinates as phi approaches 0?
  5. Ties togetherHow do cylindrical and spherical coordinates each simplify a different quadric surface's equation?
Your turn1 of 2
Find rho, the spherical radial distance, for the Cartesian point (2, 2, 5).

2.7 Converting Between Cartesian, Cylindrical, and Spherical Coordinates

  1. Worked exampleFind the cylindrical and spherical coordinates of the Cartesian point (2, -2, 1).
  2. Worked exampleFind the cylindrical coordinates of the spherical point (4, π/2, π/6).
  3. Going furtherWhat happens to a point's spherical coordinates once it lies exactly on the z-axis?
  4. Common mistakeWhy doesn't a point's cylindrical radius r equal its spherical radius rho, except when the point lies in the xy-plane?
Your turn1 of 2
Find the spherical radial coordinate rho for the Cartesian point (2, 2, 5).

2.1 Finding a Sphere's Center and Radius by Completing the Square

  1. How it worksHow does completing the square in x, y, and z turn a general quadratic equation into the standard form of a sphere?
  2. Worked exampleFind the center and radius of the sphere x2 + y2 + z2 - 4x + 2y - 6z + 5 = 0.
  3. Common mistakeWhy doesn't every equation built from x2, y2, and z2 terms actually describe a real sphere?
Your turn1 of 2
Find the x-coordinate of the center of the sphere x2 + y2 + z2 + 5x + 4y + 10z + -2 = 0.

2.5 Parallel, Perpendicular, and Skew Lines in Space

  1. How it worksHow do you compare two lines' direction vectors to test whether the lines are parallel, perpendicular, or neither?
  2. Worked exampleFind whether the lines r1(t) = (1,2,0) + t(1,1,1) and r2(s) = (0,1,2) + s(2,2,2) are parallel, and if not, whether they intersect.
  3. Worked exampleShow that the lines r1(t) = (t, 1, 0) and r2(s) = (0, s, 1) are skew.
Your turn1 of 2
Line r1 has direction vector (1, 1, 1) and line r2 has direction vector (2, 2, 2). Assuming the two lines are not identical, what is their relationship?