2.4 The Cross Product

3 min read•june 24, 2024

The is a powerful tool in , creating a new vector perpendicular to two input vectors. It's used to find areas, volumes, and solve physics problems. The resulting vector's equals the area of the formed by the inputs.

Calculating cross products involves determinants and . The operation is not commutative but is over addition. Cross products are crucial for finding perpendicular vectors, computing in physics, and determining areas and volumes in geometry.

The Cross Product

Cross product calculation and interpretation

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  • Computes new vector a⃗×b⃗\vec{a} \times \vec{b} perpendicular to both input vectors a⃗\vec{a} and b⃗\vec{b}
  • Resulting vector magnitude equals area of parallelogram formed by a⃗\vec{a} and b⃗\vec{b}
  • Direction determined by
    • Index finger along a⃗\vec{a}, middle finger along b⃗\vec{b}, thumb points in a⃗×b⃗\vec{a} \times \vec{b} direction
  • Not commutative a⃗×b⃗=−(b⃗×a⃗)\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})
  • Distributive over addition a⃗×(b⃗+c⃗)=(a⃗×b⃗)+(a⃗×c⃗)\vec{a} \times (\vec{b} + \vec{c}) = (\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c})
  • The cross product is a fundamental operation in vector algebra

Determinants for cross product computation

  • Compute a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3) and b⃗=(b1,b2,b3)\vec{b} = (b_1, b_2, b_3) cross product using a⃗×b⃗=∣[i](https://www.fiveableKeyTerm:i)^[j](https://www.fiveableKeyTerm:j)^[k](https://www.fiveableKeyTerm:k)^a1a2a3b1b2b3∣\vec{a} \times \vec{b} = \begin{vmatrix} \hat{[i](https://www.fiveableKeyTerm:i)} & \hat{[j](https://www.fiveableKeyTerm:j)} & \hat{[k](https://www.fiveableKeyTerm:k)} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}
    • i^,j^,k^\hat{i}, \hat{j}, \hat{k} represent unit vectors in x, y, z directions ()
  • Expand determinant to find resulting vector components
    • (a⃗×b⃗)x=a2b3−a3b2(\vec{a} \times \vec{b})_x = a_2b_3 - a_3b_2
    • (a⃗×b⃗)y=a3b1−a1b3(\vec{a} \times \vec{b})_y = a_3b_1 - a_1b_3
    • (a⃗×b⃗)z=a1b2−a2b1(\vec{a} \times \vec{b})_z = a_1b_2 - a_2b_1

Perpendicular vectors using cross products

  • Plane defined by point PP and n⃗\vec{n} perpendicular to any vector in plane
  • Find vector perpendicular to plane defined by non-parallel vectors a⃗\vec{a} and b⃗\vec{b} by computing their cross product
    • Resulting vector perpendicular to both a⃗\vec{a}, b⃗\vec{b}, and the plane they define
  • The cross product ensures between the resulting vector and the input vectors

Areas and volumes with cross products

  • Parallelogram area formed by vectors a⃗\vec{a} and b⃗\vec{b} equals magnitude of their cross product
    • Area = ∣a⃗×b⃗∣|\vec{a} \times \vec{b}|
  • volume formed by vectors a⃗\vec{a}, b⃗\vec{b}, c⃗\vec{c} equals
    • Volume = ∣(a⃗×b⃗)⋅c⃗∣|(\vec{a} \times \vec{b}) \cdot \vec{c}|
    • Compute scalar triple product using determinant (a⃗×b⃗)⋅c⃗=∣a1a2a3b1b2b3c1c2c3∣(\vec{a} \times \vec{b}) \cdot \vec{c} = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

Cross products in torque problems

  • Torque measures turning force on object, defined as cross product of position vector r⃗\vec{r} (from rotation axis to force application point) and force vector F⃗\vec{F}
    • τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}
  • Torque magnitude ∣τ⃗∣=∣r⃗∣∣F⃗∣sin⁡θ|\vec{\tau}| = |\vec{r}||\vec{F}|\sin\theta, θ\theta is angle between r⃗\vec{r} and F⃗\vec{F}
    • Magnitude maximized when r⃗\vec{r} and F⃗\vec{F} perpendicular (θ=90∘\theta = 90^\circ)
  • Torque direction perpendicular to both r⃗\vec{r} and F⃗\vec{F}, follows right-hand rule
  • Solving torque problems
    1. Identify position vector r⃗\vec{r} and force vector F⃗\vec{F}
    2. Calculate cross product τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F} to find torque vector
    3. Use torque vector magnitude and direction to analyze system

Key Terms to Review (23)

A × b: The cross product, denoted as a × b, is a binary operation in vector algebra that produces a vector that is perpendicular to both of the input vectors, a and b. The cross product is an important concept in mathematics, physics, and various engineering applications, as it allows for the calculation of quantities such as torque, angular momentum, and the area of a parallelogram formed by the input vectors.
Angular Momentum: Angular momentum is a measure of the rotational motion of an object around a fixed axis. It describes the object's resistance to changes in its rotational state and is a conserved quantity in isolated systems. This concept is fundamental in understanding the dynamics of rotating systems in various fields, including classical mechanics, astrophysics, and quantum mechanics.
Anticommutative: Anticommutativity is a property of a binary operation where the order of the operands affects the result of the operation. In other words, the operation is not commutative, meaning that the result changes when the order of the operands is reversed.
Cartesian Coordinates: Cartesian coordinates are a system used to locate points in space by specifying their positions along orthogonal (perpendicular) axes. This coordinate system provides a way to describe the location of objects in two-dimensional (2D) or three-dimensional (3D) space using numerical values.
Cross Product: The cross product, denoted by the symbol '×', is a binary operation in vector algebra that takes two vectors and produces a third vector that is perpendicular to both of the original vectors. The cross product is a fundamental concept in 3-dimensional space and is essential for understanding and analyzing various geometric and physical phenomena.
Determinant: The determinant is a scalar value associated with a square matrix that has important applications in linear algebra, including calculating the inverse of a matrix and determining the solvability of a system of linear equations. It provides a way to quantify the size and orientation of a geometric object represented by the matrix.
Distributive: In mathematics, distributive refers to the property that allows you to multiply a single term by two or more terms inside parentheses. This property can be expressed as a(b + c) = ab + ac, meaning that the term outside the parentheses is distributed to each term inside. This concept is crucial for simplifying expressions and solving equations, especially when working with vectors and operations like the cross product.
I: The imaginary unit, denoted by the symbol 'i', is a mathematical constant that represents the square root of -1. It is a fundamental concept in complex number theory and has important applications in various areas of mathematics, including linear algebra, calculus, and quantum mechanics.
J: The term 'j' is a vector component that represents the second dimension or y-axis in a three-dimensional Cartesian coordinate system. It is one of the three orthogonal unit vectors, along with 'i' and 'k', that define the basis for describing the position and orientation of objects in three-dimensional space.
K: In the context of the cross product, 'k' represents a unit vector that points in the direction of the z-axis in a three-dimensional Cartesian coordinate system. It is one of the standard basis vectors used to express vector components and plays a crucial role in defining the orientation of vectors in space, especially when calculating the cross product of two vectors.
Lagrange's Formula: Lagrange's formula, also known as the cross product formula, is a mathematical expression that defines the cross product of two vectors in three-dimensional space. The cross product is a vector operation that produces a new vector that is perpendicular to the two input vectors, and its magnitude is equal to the product of the magnitudes of the input vectors and the sine of the angle between them.
Magnitude: Magnitude is a fundamental concept in mathematics and physics that describes the size, scale, or intensity of a quantity. It is a measure of the absolute value or strength of a vector or a scalar, and it is a crucial component in understanding and analyzing various mathematical and physical phenomena.
Newton-Meter: A Newton-meter (N⋅m) is a unit of torque, which is a measure of the rotational force that can cause an object to rotate about an axis, fulcrum, or pivot. It is the product of force (in Newtons) and the perpendicular distance (in meters) from the axis of rotation to the line of action of the force. This unit is commonly used in the context of rotational mechanics and is particularly relevant when analyzing the cross product of two vectors.
Normal Vector: A normal vector is a vector that is perpendicular or orthogonal to a given surface, curve, or plane in three-dimensional space. It is a fundamental concept in calculus, geometry, and physics, as it helps describe the orientation and properties of various geometric objects.
Orthogonality: Orthogonality is a fundamental concept in mathematics that describes the relationship between two or more vectors, functions, or other mathematical objects. It refers to the property of being perpendicular or at right angles to one another, indicating a lack of correlation or interdependence between the objects.
Parallelepiped: A parallelepiped is a three-dimensional geometric shape formed by six parallelograms. It is a generalization of the rectangular prism, where the faces are not necessarily rectangles but can be any parallelogram. The defining feature of a parallelepiped is that its faces are all parallel and congruent to each other.
Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides. It is a fundamental geometric shape with unique properties that are particularly relevant in the context of the cross product, a key operation in vector calculus.
Right-Hand Rule: The right-hand rule is a mnemonic used in mathematics and physics to determine the direction of a vector resulting from the cross product of two other vectors. By aligning the fingers of your right hand with the first vector and curling them toward the second vector, your thumb points in the direction of the resulting vector. This concept is crucial for understanding orientations in three-dimensional space and helps in visualizing rotational directions.
Scalar Triple Product: The scalar triple product, also known as the mixed product, is a mathematical operation that combines three vectors to produce a scalar value. It is a special case of the more general vector triple product, and it provides a way to measure the volume of a parallelepiped formed by three vectors.
Torque: Torque is a measure of the rotational force that causes an object to rotate about an axis, fulcrum, or pivot. It is the product of the force applied and the perpendicular distance between the line of action of the force and the axis of rotation. Torque is a crucial concept in the study of rotational motion and equilibrium.
Unit Vectors: A unit vector is a vector with a magnitude of 1 that points in a specific direction. Unit vectors are used to represent the direction of a vector without regard to its magnitude, and they play a crucial role in the context of the cross product.
Vector Algebra: Vector algebra is the branch of mathematics that deals with the operations and properties of vectors, which are mathematical objects that have both magnitude and direction. It provides a framework for representing and manipulating quantities in multi-dimensional spaces, such as those encountered in physics and engineering.
Vector Product: The vector product, also known as the cross product, is a binary operation on two vectors that results in a third vector that is perpendicular to both of the original vectors. It is a fundamental concept in the study of three-dimensional geometry and has numerous applications in physics and engineering.
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