Question Video: Subtracting Two Vectors Then Finding the Cross Product of Their Sum by a Third Vector Mathematics

Video Transcript

Given that 𝐀 is equal to 𝐢 plus 𝐣, 𝐁 is equal to three 𝐢 minus 𝐣, and 𝐂 is equal to five 𝐢 plus three 𝐣, determine the cross product of 𝐀 minus 𝐁 and 𝐂.

In this question, we are given three vectors 𝐀, 𝐁, and 𝐂 in the coordinate plane with 𝐢 and 𝐣 as unit vectors. We are asked to calculate 𝐀 minus 𝐁 and then find the cross product of this vector with vector 𝐂. In order to calculate the vector 𝐀 minus 𝐁, we subtract the vector three 𝐢 minus 𝐣 from the vector 𝐢 plus 𝐣. We can do this by subtracting the 𝐢- and 𝐣-components separately. 𝐢 minus three 𝐢 is equal to negative two 𝐢, and 𝐣 minus negative 𝐣 is equal to two 𝐣. 𝐀 minus 𝐁 is therefore equal to negative two 𝐢 plus two 𝐣.

We want to find the cross product of this vector and vector 𝐂. This is the cross product of negative two 𝐢 plus two 𝐣 and five 𝐢 plus three 𝐣. If we have two vectors 𝐌 and 𝐍 such that 𝐌 is equal to 𝑒𝐢 plus 𝑓𝐣 and 𝐍 is equal to 𝑔𝐢 plus ℎ𝐣, then the cross product of vectors 𝐌 and 𝐍 is equal to the determinant of the two-by-two matrix 𝑒, 𝑓, 𝑔, ℎ multiplied by the unit vector 𝐊, where this unit vector is perpendicular to the 𝑥𝑦-plane.

In this question, we need to find the determinant of the two-by-two matrix negative two, two, five, three and then multiply this by the unit vector 𝐊. The determinant of the matrix is equal to negative two multiplied by three minus five multiplied by two. This is equal to negative six minus 10, which is equal to negative 16. If 𝐀 is equal to 𝐢 plus 𝐣, 𝐁 is equal to three 𝐢 minus 𝐣, and 𝐂 is equal to five 𝐢 plus three 𝐣, then the cross product of 𝐀 minus 𝐁 and 𝐂 is negative 16𝐊.