91Ó°ÊÓ

To find a vector that is orthogonal to two given vectors, we use the cross product. The cross product of two vectors in three-dimensional space results in another vector that is perpendicular to both of the original vectors. This is especially useful in determining a vector that can complete a set of orthogonal vectors in \(\mathbb{R}^3\).

Given vectors \(\mathbf{a} = (1, 3, 1)\) and \(\mathbf{b} = (2, 1, 1)\), we find \(\mathbf{c}\) by computing \(\mathbf{a} \times \mathbf{b}\).

How to calculate the cross product:

• Create a 3x3 matrix with the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) in the first row, the components of \(\mathbf{a}\) in the second row, and the components of \(\mathbf{b}\) in the third row.
• Compute the determinant to get the cross product:\[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \1 & 3 & 1 \2 & 1 & 1 \end{vmatrix}.\]
• Expand this determinant:

\[\mathbf{c} = (3 \cdot 1 - 1 \cdot 1) \mathbf{i} - (1 \cdot 1 - 1 \cdot 2) \mathbf{j} + (1 \cdot 1 - 3 \cdot 2) \mathbf{k}.\]
This simplifies to \(\mathbf{c} = (2, 1, -5)\), which is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\).

In the context of vectors, linear independence is a key concept. Three vectors in \(\mathbb{R}^3\) are considered linearly independent if no vector can be expressed as a linear combination of the others.

For our vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\), this means that there are no scalar values \(x\), \(y\), and \(z\), not all zero, such that:

• \(x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}\).

Linear independence is crucial for our set of vectors to qualify as a basis.

Determining linear independence:

• Arrange \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) as a matrix and calculate its determinant:

\[\begin{vmatrix}1 & 3 & 1 \2 & 1 & 1 \2 & 1 & -5\end{vmatrix} = -6 + 36 = 30\]
If the determinant is non-zero, the vectors are linearly independent.
This result confirms our set is linearly independent and qualifies as a basis for \(\mathbb{R}^3\).

A basis for \(\mathbb{R}^3\) is a set of three vectors that are linearly independent and span the entire \(\mathbb{R}^3\) space.

This implies any vector in \(\mathbb{R}^3\) can be expressed as a linear combination of the basis vectors. For a set of vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) to form a basis, they must be linearly independent and span \(\mathbb{R}^3\).

Characteristics of a basis:

• Three vectors are needed in three-dimensional space.
• The vectors must be linearly independent.
• The vectors must span the whole space, meaning any vector in the space can be formed by combining the basis vectors.

Since \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c} = (2, 1, -5)\) are linearly independent and, by verification, orthogonal, they indeed form a basis for \(\mathbb{R}^3\). This ensures any vector in \(\mathbb{R}^3\) can be reconstructed from these basis vectors.