Difficulty: Intermediate | Prerequisites: §12.2 (vectors), §12.3 (dot product and projections), determinants of 2x2 and 3x3 matrices.
The cross product is the companion operation to the dot product. Where the dot product measures alignment (how parallel two vectors are), the cross product produces a new vector that is perpendicular to both inputs. This is the tool you need whenever you want a normal direction to a surface, the area of a parallelogram, or a way to build an orthonormal coordinate frame. It underpins everything from torque in physics to surface integrals later in the course.
The cross product of two vectors in R³ gives a third vector perpendicular to both, with magnitude equal to the area of the parallelogram they span. Combined with the dot product, it lets you decompose any vector in terms of three mutually perpendicular unit vectors (an orthonormal basis), which is one of the most broadly useful ideas in mathematics and science.
Cross product (vector product)
For u = ⟨u₁, u₂, u₃⟩ and v = ⟨v₁, v₂, v₃⟩, the cross product u × v is a vector perpendicular to both u and v, computed via a 3x3 determinant.
Think of it as the operation that answers: "Which direction is perpendicular to both of these vectors, and how much area do they enclose?"
Right-hand rule
The direction of u × v is determined by curling the fingers of the right hand from u toward v; the thumb points in the direction of u × v.
Orthonormal basis
A set of mutually perpendicular unit vectors that span the space. In R³, any three vectors {v₁, v₂, v₃} that are pairwise orthogonal and each have magnitude 1 form an orthonormal basis.
In simple terms, it is a coordinate system built from three perfectly perpendicular rulers, each exactly one unit long.
Orthonormal decomposition
Any vector v in R³ can be written as v = (v · v₁)v₁ + (v · v₂)v₂ + (v · v₃)v₃ when {v₁, v₂, v₃} is an orthonormal basis. Each coefficient (v · vᵢ) is the component of v along vᵢ.
Distance from a point to a line
Given a line through points A and B, and a point C not on the line, the perpendicular distance from C to the line is |AB × AC| / |AB|.
The cross product u × v is evaluated as the determinant of a 3x3 matrix with i, j, k in the first row, u's components in the second, and v's components in the third.
Expand along the first row using cofactors:
i-component: u₂v₃ - u₃v₂
j-component: -(u₁v₃ - u₃v₁)
k-component: u₁v₂ - u₂v₁
The result is a vector, not a scalar.
Anti-commutative: u × v = -(v × u). Swapping the order reverses the direction.
Not associative: u × (v × w) ≠ (u × v) × w in general.
Distributive: u × (v + w) = u × v + u × w.
Parallel vectors: u × v = 0 if and only if u and v are parallel (or one is the zero vector).
Magnitude: |u × v| = |u||v|sin θ, which equals the area of the parallelogram formed by u and v.
In R², there are exactly two unit vectors orthogonal to any given non-zero vector. If v = ⟨a, b⟩, then ⟨-b, a⟩/|v| and ⟨b, -a⟩/|v| are the two options.
In R³, given one vector, there are infinitely many unit vectors orthogonal to it (they form a circle on the unit sphere).
Given two non-parallel vectors in R³, exactly two unit vectors are orthogonal to both: ±(u × v)/|u × v|.
Three mutually orthogonal unit vectors {v₁, v₂, v₃} form an orthonormal basis for R³.
Any vector v can then be decomposed as v = (v · v₁)v₁ + (v · v₂)v₂ + (v · v₃)v₃.
For unit vectors, proj_{v₁} v simplifies to (v · v₁)v₁ (no division needed, since v₁ · v₁ = 1).
To build the third vector v₃ given two orthonormal vectors v₁ and v₂: compute v₁ × v₂ (or v₂ × v₁) and, if needed, choose the sign so that a specified component is positive.
The distance from point C to the line through A and B is the height of the parallelogram with sides AB and AC, divided by the base length |AB|.
Formula: d = |AB × AC| / |AB|.
This works because |AB × AC| = |AB||AC|sin θ, and |AC|sin θ is exactly the perpendicular distance.
\vec{u} \times \vec{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}|\vec{u} \times \vec{v}| = |\vec{u}|\,|\vec{v}|\sin\theta\vec{v} = (\vec{v} \cdot \vec{v}_1)\,\vec{v}_1 + (\vec{v} \cdot \vec{v}_2)\,\vec{v}_2 + (\vec{v} \cdot \vec{v}_3)\,\vec{v}_3 \quad \text{(orthonormal decomposition)}d(C, L) = \frac{|\overrightarrow{AB} \times \overrightarrow{AC}|}{|\overrightarrow{AB}|}Torque in mechanics is defined as the cross product of the position vector and the force vector, which is why a longer wrench gives more torque. In electromagnetism, the magnetic force on a moving charge is a cross product (F = qv × B). Orthonormal bases underpin signal processing, data compression, and machine learning: any time you rotate, project, or decompose data in multiple dimensions, you are using this machinery.
The cross product is not commutative. u × v and v × u point in opposite directions. Swapping the order introduces a minus sign.
The cross product is defined only in R³ (and R⁷, though that is not relevant here). There is no cross product in R².
Students sometimes confuse the magnitude formula: the dot product uses cos θ, the cross product magnitude uses sin θ. Mixing them up gives wrong answers.
A zero cross product means the vectors are parallel, not perpendicular. A zero dot product means perpendicular.
⚠️ Computing the cross product via the 3x3 determinant is a core skill. Sign errors on the j-component (remember the minus sign) are extremely common on exams.
⚠️ The distance-from-point-to-line formula is a classic exam proof question. Understand why |AB × AC|/|AB| works geometrically.
⚠️ Verifying that vectors are orthonormal (checking both unit length and pairwise orthogonality) is tested frequently. Show all dot products explicitly.
⚠️ The orthonormal decomposition v = (v · v₁)v₁ + (v · v₂)v₂ + (v · v₃)v₃ may appear as either a computation or a proof. Know both how to use it and why it works.
True or false: u × v is a scalar.
Fill in the blank: if u × v = 0 and neither vector is zero, then u and v are ______.
True or false: |u × v| gives the area of the triangle formed by u and v.
Fill in the blank: for an orthonormal basis {v₁, v₂, v₃}, the projection of v onto v₁ simplifies to ______.
True or false: i × j = k.
Answers: 1. False (it is a vector). 2. Parallel. 3. False (it gives the area of the parallelogram; the triangle is half that). 4. (v · v₁)v₁. 5. True.
Q: Find two unit vectors orthogonal to both v = ⟨-1, 2, 4⟩ and u = ⟨6, 2, 2⟩.
A: Compute v × u using the determinant: i(2·2 - 4·2) - j((-1)·2 - 4·6) + k((-1)·2 - 2·6) = i(4 - 8) - j(-2 - 24) + k(-2 - 12) = ⟨-4, 26, -14⟩. Magnitude: √(16 + 676 + 196) = √888 = 2√222. The two unit vectors are ±⟨-4, 26, -14⟩ / (2√222) = ±⟨-2, 13, -7⟩ / √222. These are the only two.
Q: Let v₁ = ⟨1/3, 2/3, -2/3⟩ and v₂ = ⟨4/(3√2), -1/(3√2), 1/(3√2)⟩. Verify that they are orthogonal unit vectors.
A: |v₁| = √(1/9 + 4/9 + 4/9) = √(9/9) = 1. |v₂| = √(16/18 + 1/18 + 1/18) = √(18/18) = 1. v₁ · v₂ = 4/(9√2) + (-2)/(9√2) + (-2)/(9√2) = 0/(9√2) = 0. Confirmed: both are unit vectors and they are orthogonal.
Q: Show that the distance from point C to a line through A and B is |AB × AC| / |AB|.
A: The area of the parallelogram with sides AB and AC is |AB × AC|. That same parallelogram has base |AB| and height h, where h is the perpendicular distance from C to the line. So |AB × AC| = |AB| · h, giving h = |AB × AC| / |AB|.
Q: Let v = ⟨1, 2, 3⟩ and {v₁, v₂, v₃} be an orthonormal basis. Verify that v = (v · v₁)v₁ + (v · v₂)v₂ + (v · v₃)v₃ using v₁ = ⟨1/3, 2/3, -2/3⟩.
A: v · v₁ = 1/3 + 4/3 - 6/3 = -1/3. So the v₁-component contributes (-1/3)⟨1/3, 2/3, -2/3⟩ = ⟨-1/9, -2/9, 2/9⟩. The remaining components along v₂ and v₃ must sum to v minus this contribution. Computing all three and adding confirms v is recovered exactly.
The cross product is essential for §12.5 (equations of planes), where the normal vector to a plane is found via a cross product of two vectors lying in the plane. It returns in Chapter 13 (vector functions and space curves) for computing the unit normal and binormal vectors, and throughout Chapter 16 (surface integrals and Stokes' theorem). The orthonormal decomposition idea generalises to Fourier analysis, principal component analysis, and virtually any area that works with high-dimensional data.
cross product, vector product, determinant method, right-hand rule, anti-commutative, area of parallelogram, orthonormal basis, orthogonal unit vectors, orthonormal decomposition, Gram-Schmidt, distance point to line, torque, normal vector, MATH 215, Calculus III, §12.4, multivariable calculus