← Math 21a: Multivariable Calculus

Unit 1: Vectors and the Geometry of Space

Points, vectors, and the two products that measure everything between them: angles, areas, and volumes.

14 topics · 50 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.

1.1 Coordinates and Distance in Three Dimensions

  1. How it worksHow does the distance formula in three dimensions come from applying the Pythagorean theorem twice in a row?
  2. Worked exampleFind the distance between the points (1, 2, 3) and (4, 6, 15).
  3. Going furtherWhat happens to the distance formula when the two points share the same x and y coordinates?
Your turn1 of 2
Find the distance between the points (2, 2, 5) and (-3, -5, -4).

1.2 Vectors: Magnitude and Direction

  1. How it worksHow do you read a vector's magnitude and direction straight off its component form?
  2. Common mistakeWhy isn't a vector the same thing as the arrow drawn on paper to represent it?
  3. Worked exampleFind the magnitude of the vector v = <3, -4, 12>.
  4. Going furtherWhat happens to a vector's components when you slide its tail to a new starting point without turning it?
Your turn1 of 2
Find the magnitude of the vector v = <9, -6, 11>.

1.3 Vector Addition and Scalar Multiplication

  1. How it worksHow does the parallelogram rule turn two vectors placed head to tail into their sum?
  2. Worked exampleFind the sum of u = <2, 5, -1> and v = <-3, 1, 4>.
  3. Common mistakeWhy doesn't multiplying a vector by negative two just shrink it?
Your turn1 of 2
Let u = <3, 2, 7> and v = <-4, -7, -5>. Find the x-component of 2u - 3v.

1.4 The Dot Product

  1. How it worksHow does the dot product turn two vectors' components into a single measure of how aligned they are?
  2. Worked exampleFind the dot product of u = <1, 2, 3> and v = <4, -5, 6>.
  3. Common mistakeWhy doesn't a dot product of zero mean one of the two vectors must be zero?
  4. Going furtherWhat happens to the dot product of two fixed-length vectors as the angle between them grows from 0 to 180 degrees?
Your turn1 of 2
Find the dot product of u = <3, 2, 7> and v = <-4, -7, -5>.

1.5 The Angle Between Two Vectors

  1. How it worksHow does solving cos(theta) equals u dot v over the product of the lengths recover the angle between two vectors?
  2. Worked exampleFind the angle in degrees between u = <1, 0, 1> and v = <0, 1, 1>.
  3. Going furtherWhat happens to the angle formula when u and v point in exactly opposite directions?
Your turn1 of 2
Find the angle in degrees between u = <7, 6, -2> and v = <3, -8, 3>.

1.6 Scalar and Vector Projections

  1. How it worksHow does the projection formula turn a dot product into a length measured along a chosen direction?
  2. Worked exampleFind the scalar projection of b = <3, 4> onto a = <1, 0>.
  3. Worked exampleFind the vector projection of b = <2, 2, 1> onto a = <1, 0, 0>.
  4. Common mistakeWhy isn't the projection of b onto a the same vector as the projection of a onto b?
Your turn1 of 2
Find the scalar projection of b = <8, -4> onto a = <7, 6>.

1.7 The Cross Product

  1. How it worksHow does the determinant formula for u cross v build a vector perpendicular to both u and v?
  2. Common mistakeWhy isn't u cross v equal to v cross u?
  3. Worked exampleFind the cross product of u = <1, 0, 0> and v = <0, 1, 0>.
  4. Going furtherWhat happens to the cross product of two vectors as the angle between them shrinks toward zero?
Your turn1 of 2
Find the z-component of u x v, where u = <4, 3, 8> and v = <-4, -8, -5>.

1.8 Areas via the Cross Product

  1. How it worksHow does the magnitude of u cross v turn into the base-times-height area of the parallelogram u and v span?
  2. Worked exampleFind the area of the parallelogram spanned by u = <2, 0, 0> and v = <1, 3, 0>.
  3. Common mistakeWhy isn't the area of a parallelogram just the product of the two side lengths, ignoring the angle between them?
Your turn1 of 2
Find the area of the parallelogram spanned by u = (7, 0, 0) and v = (3, 8, 0).

1.9 The Scalar Triple Product and Volume

  1. How it worksHow does u dot (v cross w) turn three edge vectors into the volume of the parallelepiped they span?
  2. Worked exampleFind the volume of the parallelepiped spanned by u = <1, 0, 0>, v = <0, 2, 0>, and w = <1, 1, 3>.
  3. Common mistakeWhy doesn't a scalar triple product of zero mean that one of the three vectors must itself be the zero vector?
  4. Going furtherWhat happens to the scalar triple product when all three vectors happen to lie in the same plane?
Your turn1 of 2
Find the volume of the parallelepiped spanned by u = (6, 0, 0), v = (0, 6, 0), and w = (7, -4, 1).

1.10 Equations of Lines in Space

  1. How it worksHow do a point and a direction vector combine into the parametric equations of a line?
  2. Common mistakeWhy isn't the direction vector for a given line the only vector that could describe it?
  3. Worked exampleFind parametric equations for the line through (1, 2, 3) with direction vector <2, -1, 4>.
Your turn1 of 2
A line passes through (2, 2, 5) with direction vector <-3, 1, -4>. Write its y-coordinate as a function of t.

1.11 Equations of Planes in Space

  1. How it worksHow do a normal vector and one point on a plane combine into the plane's equation?
  2. Worked exampleFind the equation of the plane through (1, 1, 1) with normal vector <2, -1, 3>.
  3. Common mistakeWhy isn't the equation of a plane the same thing as its normal vector?
  4. Going furtherWhat happens to a plane's equation when you build it from three given points instead of a point and a normal?
Your turn1 of 2
A plane passes through (2, 2, 5) with normal vector <-3, -5, 2>. Write its z-coordinate as a function of x and y.

1.12 Distance from a Point to a Line or Plane

  1. How it worksHow does projecting onto a plane's normal direction turn into the distance from a point to that plane?
  2. Worked exampleFind the distance from the point (1, 2, 3) to the plane x + 2y + 2z = 4.
  3. Worked exampleFind the distance from the point (0, 0, 0) to the line through (1, 0, 0) with direction vector <0, 1, 1>.
  4. Ties togetherHow do the cross product and the projection formula both reappear in measuring the distance from a point to a line?
Your turn1 of 2
Find the distance from the point (-5, -4, 2) to the plane 4x + 2y + 5z = -4.

1.13 Spheres and Completing the Square

  1. How it worksHow does completing the square in each variable turn a messy 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 containing x2, y2, and z2 terms automatically describe a sphere?
Your turn1 of 2
Find the radius of the sphere x2 + y2 + z2 + 3x + 2y + 7z + -9 = 0.

1.14 Parallel and Perpendicular Lines and Planes

  1. How it worksHow does comparing two direction vectors, or two normal vectors, decide whether two lines or two planes are parallel or perpendicular?
  2. Worked exampleShow whether the planes x + 2y - z = 3 and 2x + 4y - 2z = 7 are parallel, and whether they are the exact same plane.
  3. Common mistakeWhy doesn't having proportional normal vectors automatically mean two planes are the exact same plane?
  4. Ties togetherHow does checking parallel and perpendicular lines and planes reuse the dot product and cross product?
Your turn1 of 2
Find the angle in degrees between the line with direction vector <7, 6, -2> and the line with direction vector <3, -8, 3>.