Source: MATH 163 Calculus III, University of Washington
Difficulty: Intermediate | Prerequisites: Dot product (Section 12.3), standard basis vectors in R3, basic matrix notation.
The cross product is a way of multiplying two vectors in three-dimensional space that produces a new vector, rather than a scalar. It sits alongside the dot product as one of the two fundamental vector operations in Calculus III, and you will use it constantly when working with planes, surface normals, and torque. To follow this material comfortably, you should already be able to compute dot products and know what the standard basis vectors i, j, k represent.
The cross product of two vectors gives a third vector that is perpendicular to both, with a magnitude equal to the area of the parallelogram they span. You can compute it using the component formula or, equivalently, by evaluating a 3x3 determinant. The scalar triple product extends this idea to three vectors and gives the volume of a parallelepiped.
Cross product (vector product)
An operation on two vectors in R3 that returns a vector perpendicular to both, with magnitude ||v|| ||w|| sin(theta). In simple terms, it answers: "What direction is perpendicular to these two vectors, and how big is the parallelogram they make?"
Normal vector
A unit vector n that is perpendicular to both input vectors, chosen by the right-hand rule. Think of it as the direction the cross product "points."
Determinant (2x2)
For a matrix [[a, b], [c, d]], the determinant equals ad - bc. In simple terms, it is a single number you get by multiplying diagonally and subtracting.
Determinant (3x3)
Computed by cofactor expansion along the first row: a(ei - fh) - b(di - fg) + c(dh - eg). This is the tool that makes cross product calculations systematic.
Scalar triple product
The dot product of one vector with the cross product of two others: r . (v x w). It equals the determinant of the 3x3 matrix whose rows are the three vectors, and it gives the volume of the parallelepiped they span.
Parallelepiped
A three-dimensional figure formed by three vectors, like a slanted box. Think of it as the 3D version of a parallelogram.
Right-hand rule
A physical mnemonic: curl the fingers of your right hand from v toward w, and your thumb points in the direction of v x w. This is how the direction of the cross product is determined.
The cross product of v and w is defined as: v x w = ||v|| ||w|| sin(theta) n, where theta is the angle between the two vectors and n is the unit normal vector determined by the right-hand rule.
The result is a vector, not a scalar. This is why it is also called the "vector product."
The cross product only works in R3. The dot product works in any dimension.
These results follow directly from the definition and the right-hand rule. You should memorise them:
i x j = k
j x k = i
k x i = j
j x i = -k (reversing the order flips the sign)
i x i = 0 (any vector crossed with itself gives the zero vector)
The cyclic pattern (i, j, k, i, j, k...) going forward gives a positive result. Going backward gives a negative result.
v x v = 0 (a vector crossed with itself is always the zero vector, because sin(0) = 0)
v x w = -(w x v) (anti-commutativity: swapping the order negates the result)
c(v x w) = (cv) x w = v x (cw) (scalars factor out freely)
v x (l + w) = (v x l) + (v x w) (distributive over addition)
Given v = (v_x, v_y, v_z) and w = (w_x, w_y, w_z):
v x w = (v_y w_z - v_z w_y, v_z w_x - v_x w_z, v_x w_y - v_y w_x)
Each component is a "mini" 2x2 determinant formed by covering one row and one column of a matrix.
Worked example: v = (1, -1, 5) and w = (-3, 2, 4) gives v x w = (-4 - 10, -15 - 4, 2 - 3) = (-14, -19, -1).
The component formula is equivalent to evaluating the determinant of a 3x3 matrix with the basis vectors in the first row, the components of v in the second row, and the components of w in the third row.
For a 2x2 determinant: |a b; c d| = ad - bc.
For a 3x3 determinant, expand along the first row (cofactor expansion): a(ei - fh) - b(di - fg) + c(dh - eg).
The cross product as a determinant: v x w = |i j k; v_x v_y v_z; w_x w_y w_z|.
This is the standard method for computing cross products in practice.
Class example: (2, 7, -4) x (5, -2, 1) = (-1, -22, -39).
The magnitude ||v x w|| equals the area of the parallelogram formed by v and w. This follows directly from the formula: ||v|| ||w|| sin(theta) is the base times height of that parallelogram.
The scalar triple product r . (v x w) equals the determinant of the 3x3 matrix with rows r, v, w. Its absolute value gives the volume of the parallelepiped spanned by the three vectors.
If the scalar triple product equals zero, the three vectors are coplanar (they all lie in the same plane).
\vec{v} \times \vec{w} = \|\vec{v}\| \, \|\vec{w}\| \sin(\theta) \, \hat{n}\vec{v} \times \vec{w} = \langle v_y w_z - v_z w_y,\; v_z w_x - v_x w_z,\; v_x w_y - v_y w_x \rangle\vec{v} \times \vec{w} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix}\vec{r} \cdot (\vec{v} \times \vec{w}) = \begin{vmatrix} r_x & r_y & r_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{vmatrix}\|\vec{v} \times \vec{w}\| = \text{Area of the parallelogram spanned by } \vec{v} \text{ and } \vec{w}Torque in physics is defined as r x F, the cross product of the position vector and the force vector. This is why a longer wrench makes it easier to loosen a bolt: the magnitude of the cross product increases with the length of r.
In computer graphics, cross products are used to find the normal vector to a surface, which determines how light bounces off that surface. Every shaded 3D object on your screen relies on cross products being computed millions of times per frame.
Students often assume the cross product is commutative, like the dot product. It is not. Swapping the order of the two vectors negates the result: v x w = -(w x v).
The cross product produces a vector, not a scalar. If you get a single number, you have likely confused it with the dot product.
The cross product is only defined in R3. Trying to take the cross product of two 2D vectors without adding a third component (z = 0) will cause errors.
When using the determinant method, a common sign error is forgetting the alternating signs in cofactor expansion: the j-component gets a minus sign in front, not a plus.
Computing cross products via the determinant method is almost guaranteed to appear on the final. Practise until you can do it without looking up the formula.
The geometric interpretation (area of the parallelogram) appears frequently as an application problem. You may be given two vectors and asked for the area of the triangle they span, which is half the magnitude of the cross product.
Scalar triple product questions typically ask you to find the volume of a parallelepiped or to determine whether three vectors are coplanar.
Know the properties cold, especially anti-commutativity and the fact that v x v = 0. These appear as true/false or short-answer questions.
True or false: v x w = w x v.
Fill in the blank: i x j = ___.
True or false: the cross product of two parallel vectors is the zero vector.
Fill in the blank: the magnitude of v x w equals the area of the ___ formed by v and w.
True or false: the cross product is defined in R2.
Answers: 1. False (it equals -(w x v)). 2. k. 3. True. 4. Parallelogram. 5. False (R3 only).
Q: Compute (1, -1, 5) x (-3, 2, 4).
A: (-4 - 10, -15 - 4, 2 - (-3)) = (-14, -19, -1).
Q: Compute (2, 7, -4) x (5, -2, 1).
A: (7 - 8, -20 - 2, -4 - 35) = (-1, -22, -39).
Q: Find the area of the parallelogram spanned by v = (1, 0, 3) and w = (2, 1, 0).
A: First compute v x w = (0 - 3, 6 - 0, 1 - 0) = (-3, 6, 1). Then ||(-3, 6, 1)|| = sqrt(9 + 36 + 1) = sqrt(46). The area is sqrt(46).
Q: Three vectors r = (1, 0, 0), v = (0, 1, 0), w = (0, 0, 1). Find the volume of the parallelepiped they span.
A: The scalar triple product is the determinant of the 3x3 identity matrix, which equals 1. The volume is |1| = 1.
Q: Are the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9) coplanar?
A: Compute the scalar triple product (the 3x3 determinant). It equals 0, so yes, they are coplanar.
The cross product is essential for defining planes in R3 (Section 12.5), because the normal vector to a plane is found via a cross product of two direction vectors lying in that plane. It also connects to torque and angular momentum in physics courses, and to surface integrals and curl in later sections of Calculus III. If you are comfortable with cross products, the transition to planes and surfaces will be considerably smoother.
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