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16.4: Cross Product

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    The cross product is a mathematical operation that can be performed on any two three-dimensional vectors. The result of the cross product operation will be a third vector that is perpendicular to both of the original vectors and has a magnitude of the first vector times the magnitude of the second vector times the sine of the angle between the vectors.

    Two vectors A and B lie in the plane of the "ground". Their tails are located at the same point, making an angle of theta between them, and vector A points to the left and into the screen while vector B points leftward and slightly out of the screen. The cross product of A and B is a third vector C that points upward, in the plane of the "wall" that is perpendicular to the plane of the "ground," from the point of intersection of A and B.
    Figure \(\PageIndex{1}\): The cross product of two vectors will be a vector that is perpendicular to both original vectors with a magnitude of \(|\vec{A}|\) times \(|\vec{B}|\) times the sine of the angle between \(\vec{A}\) and \(\vec{B}\).

    When finding a cross product, you may notice that there are actually two directions that are perpendicular to both of your original vectors. These two directions will be in exact opposite directions. To find which of these two directions the cross product uses, we will use the right-hand rule.

    To use the right-hand rule, hold out you right hand, point your index finger in the direction of the first vector, turn your middle finger in towards the direction of the second vector, and hold your thumb up. Your thumb should now point in the direction of the cross product vector.

    The set of vectors from Figure 1 above are shown, as well as a person's right hand. The hand's extended index finger points in the direction of vector A: leftward and into the page. The extended middle finger points in the direction of vector B: leftwards and slightly out of the page. The extended thumb points upwards, in the direction of vector C, or the product of A cross B.
    Figure \(\PageIndex{2}\): We can use the right-hand rule to determine the direction of the cross product. Image adapted from work by Acdx, license CC-BY-SA 3.0.

    One additional thing you can note with the right-hand rule is that switching the order of the two input vectors (switching \(\vec{A}\) and \(\vec{B}\)) would result in the cross product pointing in exactly the opposite direction. This is because the cross product operation is not commutative, meaning that order does matter. Specifically, switching the order of the inputs gives you a result that is exactly the opposite of what your original calculation.

    Calculating the Cross Product

    To find the cross product by hand, the easiest method is as follows.

    1. Write out the letters \(x \,\, y \,\, z \,\, x \,\, y\) in a row as shown in the diagram below.
    2. Write out the \(x\), \(y\), and \(z\) components of the first vector underneath the corresponding letters of the row of letters from Step 1. Repeat this for the second vector, writing out the second vector's components in a row under the first vector.
    3. Draw in diagonals as shown in the diagram. The diagonals that travel to the right as they move down represent positive quantities, while the diagonals that travel to the left as they move down represent negative quantities.
    4. Using the letters the diagonals travel through in the top row as a guide for which component of the result each quantity is part of, take the sum of the positive and negative diagonal products for each of the three components in the result. This should give you the final formula shown in the diagram.
    Vector A contains components [a, b, c], and Vector B contains components [d, e, f]. The letters x, y, z, x, y are written out in a horizontal row. In a second row, the components of A are written below their corresponding variables: a, b, c, a, b. In a third row, the components of B are written in the order d, e, f, d, e. Three diagonal lines run through the matrix, moving from the upper left to the lower right: one passes through x, b, f; another through y, c, d; the third through z, a, e. Moving from the upper right to the lower left, three other diagonal lines run through the matrix: one passes through y, a, f; another through x, c, e; the third through z, b, d. At the bottom of the diagram, the answer for the cross product of vector A and vector B is given in component form [(bf-ce), (cd-af), (ae-bd)].
    Figure \(\PageIndex{3}\): We can use the visual aid above to help remember the formula for the cross product.

    In addition to calculating the cross product by hand, we can also use computer tools such as the "cross" command in MATLAB or web-based tools such as the Wolfram Alpha Vector Operation calculator. Access to these tools allows you to very easily and quickly calculate the cross product and is a major advantage to using vectors operations to analyze problems.

    Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/S0eav05be_U.

    Example \(\PageIndex{1}\)

    Calculate the cross product of force vectors \(\vec{A}\) and \(\vec{B}\) in the diagram below by hand.

    A standard-orientation Cartesian coordinate plane, with two force vectors radiating out from the origin. Vector A has a magnitude of 5 kN and points in the positive x-direction. Vector B has a magnitude of 3 kN and points 30° clockwise from the negative y-direction.
    Figure \(\PageIndex{4}\): problem diagram for Example \(\PageIndex{1}\). Two force vectors radiate out from the origin of a Cartesian coordinate plane.
    Solution:
    Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/KQhOkEflq9s.

    Example \(\PageIndex{2}\)

    Calculate the cross product of the vectors \(\vec{A}\) and \(\vec{B}\) in the diagram below by hand.

    Vector A has x, y, z components [1, 3, 5]. Vector B has x, y, z components [6, 4, 2]. What is their cross product?
    Figure \(\PageIndex{5}\): problem diagram for Example \(\PageIndex{2}\).
    Solution:
    Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/oHLU_q3kVKc.

    Example \(\PageIndex{3}\)

    Calculate the cross product of the vectors \(\vec{A}\) and \(\vec{B}\) in the diagram below using MATLAB.

    Vector A has x, y, z components [1, 3, 5]. Vector B has x, y, z components [6, 4, 2]. What is their cross product?

    Figure \(\PageIndex{6}\): problem diagram for Example \(\PageIndex{3}\).

    Solution:
    Video \(\PageIndex{4}\): Worked solution to example problem \(\PageIndex{3}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/a9w6GWQJ96A.

    Example \(\PageIndex{4}\)

    Calculate the cross product of vectors \(\vec{A}\) and \(\vec{B}\) in the diagram below, using the Wolfram Vector Operation Calculator.

    Vector A has x, y, z components [1, 3, 5]. Vector B has x, y, z components [6, 4, 2]. What is their cross product?
    Figure \(\PageIndex{7}\): problem diagram for Example \(\PageIndex{4}\).
    Solution:
    Video \(\PageIndex{5}\): Worked solution to example problem \(\PageIndex{4}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/24JCFHWWGG4.

    This page titled 16.4: Cross Product is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform.