Difficulty: Introductory-Intermediate | Prerequisites: basic trigonometry, Calculus I and II
Vectors are one of the foundational tools of multivariable calculus. This material sits right at the start of the course and underpins everything that follows, from parametric curves to surface normals. You should already be comfortable with coordinate geometry in 2D and basic trig identities.
Vectors in 3D let you describe direction and magnitude in space. The dot product tells you about angles between vectors (and gives a scalar), the cross product gives a new vector perpendicular to both inputs (and its magnitude encodes area), and the scalar triple product measures volume. If vectors form a closed loop (like the sides of a triangle traversed in order), they sum to zero.
Dot product (scalar product)
For vectors u = ⟨a, b, c⟩ and v = ⟨d, e, f⟩, the dot product is u · v = ad + be + cf. It equals |u||v|cosθ, where θ is the angle between them. The result is a scalar, not a vector.
In simple terms, this measures how much two vectors point in the same direction.
Cross product (vector product)
For vectors u = ⟨a, b, c⟩ and v = ⟨d, e, f⟩, the cross product u × v = ⟨bf − ce, cd − af, ae − bd⟩. The result is a vector perpendicular to both u and v.
Think of it as: the cross product builds a new vector that sticks straight out of the plane containing the two inputs.
Magnitude of the cross product
|u × v| = |u||v|sinθ. This equals the area of the parallelogram formed by u and v.
In simple terms, it tells you how much "spread" there is between the two vectors.
Scalar triple product
(u × v) · w. This gives a scalar whose absolute value equals the volume of the parallelepiped formed by u, v, and w. If the three vectors are coplanar, the scalar triple product is zero.
Think of it as: if three vectors all lie in the same plane, the "box" they form is flat, so volume = 0.
Closed-loop vector sum
When vectors represent the sides of a polygon traversed in order (each vector's tail starts at the previous vector's head, ending back at the start), they sum to the zero vector. For a triangle with vertices P₁, P₂, P₃: the vector from P₁→P₂, plus the vector P₂→P₃, plus the vector P₃→P₁ = 0.
If vectors u, v, w form a closed traversal of a triangle (each starts where the last ended), then u + v + w = 0.
This holds regardless of the triangle's shape, size, or position in space. No coordinates are needed to know this.
Example: for an equilateral triangle with vertices P₁, P₂, P₃, define u = P₁P₂, v = P₂P₃, w = P₃P₁. Then u + v + w = 0 because the path returns to its starting point.
For an equilateral triangle of side length s, every pair of side-vectors has the same lengths (|u| = |v| = s) but the angle between them depends on which two you pick.
The angle between consecutive side-vectors (e.g. u and v) is 120°, not 60°. The interior angle of the triangle is 60°, but the vectors are laid tip-to-tail, so the angle between their directions is 180° − 60° = 120°.
Therefore: u · v = |u||v|cos(120°) = 6 × 6 × (−1/2) = −18.
u × v produces a vector perpendicular to both u and v. Its direction depends on the specific coordinates of the vectors.
For an equilateral triangle described only by side length (no coordinates given), you know the magnitude of the cross product but not its direction.
So u × v = N/A (not enough information), but |u × v| can be computed.
|u × v| = |u||v|sinθ = 6 × 6 × sin(120°) = 36 × (√3/2) = 18√3.
Geometrically, this is the area of the parallelogram spanned by u and v. Half of it (9√3) is the area of the equilateral triangle.
(u × v) · w measures the volume of the parallelepiped formed by u, v, w.
Since w = −u − v (from the closed-loop property), all three vectors are coplanar: they lie in the plane of the triangle.
Three coplanar vectors always give a scalar triple product of 0.
Therefore (u × v) · w = 0.
\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a & b & c \\ d & e & f \end{vmatrix} = \langle bf - ce,\; cd - af,\; ae - bd \rangle\text{Vol} = |\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)|\text{Distance from } (a,b,c) \text{ to plane } Ax+By+Cz+D=0 \;=\; \frac{|Aa+Bb+Cc+D|}{\sqrt{A^2+B^2+C^2}}Students often use 60° as the angle between consecutive side-vectors of an equilateral triangle. The interior angle is 60°, but the vectors laid tip-to-tail make an angle of 120°. Always sketch the vectors from a common origin to see the true angle.
Students sometimes think the cross product of two vectors can always be computed from magnitudes and the angle alone. You can find the magnitude that way, but the direction requires knowing the actual coordinate components. If coordinates are not given, the cross product itself is N/A.
Students confuse dot product (scalar output) with cross product (vector output). The dot product is commutative (u · v = v · u); the cross product is anti-commutative (u × v = −(v × u)).
Students forget that three coplanar vectors always yield a scalar triple product of zero. If w can be written as a linear combination of u and v, the parallelepiped is flat.
⚠️ Expect problems where you are given geometric information (a shape, side lengths, angles) but no coordinates. You must decide which quantities can be computed and which require coordinates. "N/A" is a valid answer.
⚠️ The closed-loop vector sum (u + v + w = 0 around a triangle) appears in multiple exam questions. It also lets you express any one vector in terms of the other two, which is useful for simplifying dot products and triple products.
⚠️ The distinction between magnitude-only quantities (|u × v|, u · v) and direction-dependent quantities (u × v itself) is a recurring exam theme. Know when each formula applies.
⚠️ A correct answer with no supporting work earns only 1 point on this exam. Always show reasoning, even for results like "0" or "N/A".
True or false: the dot product of two vectors is always a vector. (False: it is a scalar.)
True or false: u × v = v × u. (False: u × v = −(v × u).)
Fill in the blank: the angle between two consecutive side-vectors of an equilateral triangle is ____. (120°.)
True or false: if three vectors are coplanar, their scalar triple product is zero. (True.)
Fill in the blank: |u × v| gives the area of the ____ spanned by u and v. (Parallelogram.)
Q: Let P₁, P₂, P₃ be vertices of an equilateral triangle of side length 6. Define u = P₁P₂, v = P₂P₃, w = P₃P₁. What is u + v + w?
A: The zero vector 0. The three vectors form a closed loop, so they sum to zero regardless of the triangle's position or orientation.
Q: Using the same setup, compute u · v.
A: u · v = |u||v|cosθ = 6 × 6 × cos(120°) = 36 × (−1/2) = −18. The angle between consecutive side-vectors is 120°, not the interior angle of 60°.
Q: Can you compute u × v given only the side length and shape? What about |u × v|?
A: u × v = N/A (you need coordinates to determine the direction of the resulting vector). However, |u × v| = 6 × 6 × sin(120°) = 18√3.
Q: Compute (u × v) · w.
A: 0. Since w = −u − v (closed-loop property), the three vectors are coplanar. The scalar triple product of coplanar vectors is always zero.
Q: Two vectors a and b have |a| = 3, |b| = 4, and the angle between them is 90°. Find |a × b| and a · b.
A: a · b = 3 × 4 × cos(90°) = 0. |a × b| = 3 × 4 × sin(90°) = 12.
The cross product reappears when you compute normal vectors to surfaces and planes (used heavily in tangent-plane problems later in the course). The scalar triple product connects to determinants, which appear in change-of-variables for multiple integrals. The dot product is the basis for projections, directional derivatives, and the gradient.
Related Terms / Search Tags: dot product, scalar product, inner product, cross product, vector product, outer product, magnitude, norm, angle between vectors, equilateral triangle vectors, scalar triple product, parallelepiped volume, coplanar vectors, closed-loop sum, MATH 215, Calculus III, multivariable calculus, University of Michigan, vector operations 3D