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

Calculate the cross product of 𝐲 = (2𝐒 + 4𝐣 + 8𝐀) and 𝐳 = (6𝐒 + 4𝐣 + 2𝐀).

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Video Transcript

Calculate the cross product of 𝐲 equals two 𝐒 plus four 𝐣 plus eight 𝐀 and 𝐳 equals six 𝐒 plus four 𝐣 plus two 𝐀.

In this example, we want to solve for 𝐲 cross 𝐳, where 𝐲 and 𝐳 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 𝐀 and 𝐁, is equal to the determinant of a three-by-three matrix , where the columns are headed by the three unit vectors 𝐒, 𝐣, and 𝐀. And the last two rows are populated by the respective components of vectors 𝐀 and 𝐁. We can apply this formula to our vectors 𝐲 and 𝐳. As we set up our matrix, we know the top row will be the 𝐒, 𝐣, and 𝐀 unit vectors. The next row will be the respective components of the vector 𝐲. Looking at 𝐲, we see that it’s π‘₯-component is two, its 𝑦-component is four, and its 𝑧-component is eight. So we write those values in to our matrix. In the next row, we’ll write the components of 𝐳. The π‘₯-component of 𝐳 is six. Its 𝑦-component is four. And its 𝑧-component is two.

Now we’re ready to calculate the determinant of this matrix and solve for the cross product of 𝐲 cross 𝐳. When we compute this determinant, we find that the 𝐒-component is eight minus 32, or negative 24. The 𝐣-component is four minus 48, or negative 44. And the 𝐀-component is eight minus 24, which equals negative 16. Our overall cross product then is negative 24𝐒 minus 44𝐣 minus 16𝐀. That’s the cross product of the vectors 𝐲 and 𝐳.

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