Video: Calculating the Cross Product of Two Vectors in Vector Notation

Calculate the cross product of π² = (2π’ + 4π£ + 8π€) and π³ = (6π’ + 4π£ + 2π€).

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Video Transcript

Calculate the cross product of π² equals two π’ plus four π£ plus eight π€ and π³ equals six π’ plus four π£ plus two π€.

In this example, we want to solve for π² cross π³, where π² and π³ are both three-dimensional vectors. We can begin our solution by recalling the mathematical definition for a cross product. The cross product of two vectors, we can call them π and π, is equal to the determinant of a three-by-three matrix , where the columns are headed by the three unit vectors π’, π£, and π€. And the last two rows are populated by the respective components of vectors π and π. We can apply this formula to our vectors π² and π³. As we set up our matrix, we know the top row will be the π’, π£, and π€ unit vectors. The next row will be the respective components of the vector π². Looking at π², we see that itβs π₯-component is two, its π¦-component is four, and its π§-component is eight. So we write those values in to our matrix. In the next row, weβll write the components of π³. The π₯-component of π³ is six. Its π¦-component is four. And its π§-component is two.

Now weβre ready to calculate the determinant of this matrix and solve for the cross product of π² cross π³. When we compute this determinant, we find that the π’-component is eight minus 32, or negative 24. The π£-component is four minus 48, or negative 44. And the π€-component is eight minus 24, which equals negative 16. Our overall cross product then is negative 24π’ minus 44π£ minus 16π€. Thatβs the cross product of the vectors π² and π³.