Difficulty: Foundational to Intermediate | Prerequisites: Basic trigonometry, coordinate geometry
Vectors are the language of multivariable calculus. Before you can talk about gradients, directional derivatives, or surface integrals, you need fluency with vector arithmetic, the dot product, and the cross product. This material sits at the very start of MATH 215 and underpins everything that follows, from parametric curves through triple integrals. If you are coming in cold, make sure you are comfortable with basic trigonometry (cosine of standard angles) and the idea of magnitude and direction in two and three dimensions.
Vectors that trace a closed path (like the sides of a triangle) always sum to the zero vector. The dot product measures how much two vectors point in the same direction, weighted by their lengths, and equals |u||v| cos(θ). The cross product gives a vector perpendicular to both inputs whose magnitude equals the area of the parallelogram they span. The scalar triple product gives the volume of the parallelepiped formed by three vectors.
Displacement vector
A vector that points from one point to another. Written as the arrow from P₁ to P₂, it encodes both the direction and the distance between the two points. In simple terms, it is the "step" you take to get from one point to another.
Dot product (scalar product)
An operation on two vectors that returns a single number: u · v = |u||v| cos(θ), where θ is the angle between them when placed tail to tail. Think of it as a measure of how aligned two vectors are. Parallel vectors give the largest magnitude; perpendicular vectors give zero.
Cross product (vector product)
An operation on two vectors in three dimensions that returns a new vector perpendicular to both. Its magnitude equals the area of the parallelogram the two vectors span: |u × v| = |u||v| sin(θ). Think of it as encoding both the "area" and the "orientation" of the plane defined by the two vectors.
Scalar triple product
The quantity a · (b × c), which returns a single number. Its absolute value equals the volume of the parallelepiped (a 3D "slanted box") determined by the three vectors. In simple terms, it tells you how much 3D space the three vectors enclose.
Closed-loop vector sum
When vectors are placed head to tail and return to the starting point (as with the sides of a triangle traversed in order), their sum is always the zero vector. Think of it as: if you walk all the way around a closed path, your net displacement is zero.
If vectors u, v, w represent the sides of a triangle traversed in order (P₁ to P₂, P₂ to P₃, P₃ to P₁), their sum is the zero vector: u + v + w = 0.
This holds for any closed polygon, not just triangles. The key is that the vectors are placed head to tail and close the loop.
This property is purely geometric; it does not depend on the shape being equilateral or on specific coordinates.
The dot product formula is u · v = |u||v| cos(θ), where θ is the angle between the vectors measured tail to tail.
In an equilateral triangle with side length 6, vectors u = P₁P₂ and v = P₂P₃ share the vertex P₂. The interior angle at P₂ is 60 degrees, but because u points into P₂ while v points away from P₂, the tail-to-tail angle is the exterior angle: 180 − 60 = 120 degrees.
Calculation: u · v = 6 × 6 × cos(120°) = 36 × (−1/2) = −18.
A negative dot product means the vectors point in "mostly opposite" directions (the angle between them exceeds 90 degrees).
|u × v| = |u||v| sin(θ), which equals the area of the parallelogram spanned by u and v.
The area of the triangle with adjacent sides u and v is half that: (1/2)|u × v|.
The direction of u × v is given by the right-hand rule: perpendicular to both u and v.
|a · (b × c)| gives the volume of the parallelepiped determined by the three vectors a, b, c.
The scalar triple product is positive when a, b, c form a right-handed system, negative when left-handed, and zero when the three vectors are coplanar.
A line segment from point A to point B can be parametrised as r(t) = A + t(B − A) = (1 − t)A + tB, for 0 ≤ t ≤ 1.
At t = 0 you get A; at t = 1 you get B.
Example: r(t) = ⟨1 − t, 2 + 3t, 4t⟩ gives (1, 2, 0) at t = 0 and (0, 5, 4) at t = 1.
\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta\text{Volume of parallelepiped} = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\text{Area of triangle} = \frac{1}{2}|\mathbf{u} \times \mathbf{v}|\mathbf{r}(t) = (1-t)\mathbf{A} + t\mathbf{B}, \quad 0 \le t \le 1The dot product is used in physics to calculate work done by a force (W = F · d), and in computer graphics to determine lighting angles on surfaces. The cross product is how engineers calculate torque (τ = r × F) and find normal vectors to surfaces for rendering and structural analysis.
Students often use the interior angle of a triangle for the dot product when the vectors share a vertex but point in "linked" directions (head to tail). The dot product formula requires the tail-to-tail angle, which is the supplement (180° minus the interior angle).
Students often confuse |u × v| (the area of the parallelogram) with the area of the triangle. The triangle is half the parallelogram.
Students sometimes think the scalar triple product gives a vector. It gives a scalar (a single number).
Students sometimes assume that u + v + w = 0 only works for equilateral triangles. It works for any closed polygon whose sides are traversed in order.
⚠️ The tail-to-tail angle versus interior angle distinction is a classic exam trap. If vectors are linked head to tail around a triangle, the angle between consecutive vectors (for dot product purposes) is the exterior angle.
⚠️ Know the geometric meaning of each product: dot product gives a scalar related to alignment, cross product gives a vector related to area, scalar triple product gives a scalar related to volume.
⚠️ Be able to verify a parametric line segment by checking the endpoints at t = 0 and t = 1.
True or false: vectors forming a closed loop always sum to zero. (True)
Fill in the blank: the dot product u · v equals zero when the vectors are ______. (perpendicular / orthogonal)
True or false: |u × v| equals the area of the triangle with sides u and v. (False, it equals the area of the parallelogram; the triangle area is half that)
Fill in the blank: the absolute value of the scalar triple product |a · (b × c)| gives the ______ of the parallelepiped. (volume)
True or false: r(t) = ⟨1 − t, 2 + 3t, 4t⟩ for 0 ≤ t ≤ 1 starts at (0, 5, 4). (False, it starts at (1, 2, 0))
Q: Let u, v, w be the sides of an equilateral triangle with side length 6, traversed in order around the triangle. What is u + v + w?
A: The zero vector, 0. Displacement vectors forming a closed loop always sum to zero.
Q: With the same triangle, calculate u · v.
A: The tail-to-tail angle between consecutive side vectors is the exterior angle, 180° − 60° = 120°. So u · v = 6 × 6 × cos(120°) = 36 × (−1/2) = −18.
Q: What is the geometric meaning of |a · (b × c)|?
A: The volume of the parallelepiped determined by the three vectors a, b, c.
Q: True or false: |u × v| equals the area of the triangle with adjacent sides u and v.
A: False. |u × v| is the area of the parallelogram. The triangle area is (1/2)|u × v|.
Q: Does the parametrisation r(t) = ⟨1 − t, 2 + 3t, 4t⟩ for 0 ≤ t ≤ 1 describe a line segment from (1, 2, 0) to (0, 5, 4)?
A: True. Substituting t = 0 gives (1, 2, 0) and t = 1 gives (0, 5, 4).
This material connects directly to gradients and directional derivatives: the gradient is a vector, and the directional derivative is computed via a dot product. Cross products reappear when computing normal vectors to surfaces for surface integrals and flux. The scalar triple product is the determinant of a 3×3 matrix, linking vector geometry to linear algebra.
vector addition, dot product, scalar product, inner product, cross product, vector product, outer product, scalar triple product, parallelepiped volume, parallelogram area, triangle area, displacement vector, closed loop, parametric line, line segment parametrisation, tail-to-tail angle, exterior angle, equilateral triangle vectors, MATH 215, Calculus III, multivariable calculus, University of Michigan