Question

Find the components of the vector sum aƒ— bƒ—?

Answer 1

The components of the vector sum
`\(\mathbf{a} * \mathbf{b}\)` can be found using the cross product formula. If
`\(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}\)`, then the components of
`\(\mathbf{a} * \mathbf{b}\)` are given by \[a_x \times b_x = a_2b_3 - a_3b_2,
`\quad a_y * b_y = a_3b_1 - a_1b_3,\]` and
`\[a_z * b_z = a_1b_2 - a_2b_1.\]`

Step-by-step explanation:

The vector sum
`\(\mathbf{a} * \mathbf{b}\)` represents the cross product of two vectors (a) and (b). The cross product is calculated using the components of the vectors and follows a specific formula. Let
`\(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\)` and
`\(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}\)`, where
`\(a_1, a_2, a_3, b_1, b_2,\)` and
`\(b_3\)` are the scalar components.

The components of the cross product
`\(\mathbf{a} * \mathbf{b}\)` are given by
`\[a_x * b_x = a_2b_3 - a_3b_2,\] \[a_y * b_y = a_3b_1 - a_1b_3,\] and \[a_z * b_z = a_1b_2 - a_2b_1.\]` These components represent the respective coefficients of the unit vectors
`\(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\)` in the resulting vector. The cross product is particularly useful in determining a vector perpendicular to the plane formed by the original vectors and is fundamental in various mathematical and physical applications.