91Ó°ÊÓ

In linear algebra, orthogonal vectors are vectors that meet at a right angle. This means their dot product is zero. The dot product is a way to multiply vectors that results in a scalar. If two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), are orthogonal, the dot product \( \mathbf{a} \cdot \mathbf{b} \) is \( 0 \).

An example: Consider vectors \( \mathbf{a} = (1, 3, 1) \) and \( \mathbf{c} = (2, -1, -5) \). We check orthogonality by computing their dot product: \( 1 \cdot 2 + 3 \cdot (-1) + 1 \cdot (-5) = 0 \). Thus, \( \mathbf{c} \) is orthogonal to \( \mathbf{a} \).

• Orthogonality is akin to the concept of 'perpendicular' in geometry.
• In a vector space, orthogonal vectors form the simplest building blocks for constructing other vectors.

The cross product is an operation on two vectors in three-dimensional space. It results in a third vector that is orthogonal to both of the original vectors. This is particularly useful when finding a vector that is perpendicular to two given vectors.

To calculate this, we use the formula involving the determinant of a 3x3 matrix. For vectors \( \mathbf{a} = (1, 3, 1) \) and \( \mathbf{b} = (2, 1, 1) \), the cross product \( \mathbf{a} \times \mathbf{b} \) can be calculated as follows:
\[ \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 3 & 1 \ 2 & 1 & 1 \end{vmatrix} \]

The result is \( \mathbf{c} = (2, -1, -5) \), which is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \).

• The direction of the cross product follows the right-hand rule.
• The cross product is only defined in three-dimensional space.

A basis for \(\mathbb{R}^3\) is a set of three vectors that span the entirety of three-dimensional space. For a set of vectors to form a basis, they must be linearly independent. This means no vector in the set can be written as a combination of the others.

In our exercise, we have vectors \( \mathbf{a} = (1, 3, 1) \), \( \mathbf{b} = (2, 1, 1) \), and \( \mathbf{c} = (2, -1, -5) \). These vectors form a basis if any point in \(\mathbb{R}^3\) can be reached by combining these vectors with scalar multipliers.

• A basis provides the framework needed to express any vector in the space using a linear combination of the basis vectors.
• The simplest example of a basis for \(\mathbb{R}^3\) involves the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).

The determinant is a special number that can be calculated from a square matrix. In the context of basis vectors, the determinant helps verify linear independence. For three vectors in \(\mathbb{R}^3\), you arrange them as a square matrix and compute its determinant. A non-zero determinant indicates that the vectors are linearly independent and thus can form a basis.

For vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) from our example, construct the matrix:
\[ \begin{vmatrix} 1 & 3 & 1 \ 2 & 1 & 1 \ 2 & -1 & -5 \end{vmatrix} \]

With a determinant value of 20, which is non-zero, these vectors are linearly independent, confirming they form a basis for \(\mathbb{R}^3\).

• The determinant can be positive or negative, indicating the orientation of the basis vectors.
• If the determinant is zero, the vectors are not independent and cannot form a basis.

Linear independence is a core concept that tells us if a set of vectors can form a basis for a vector space. Vectors are linearly independent if no vector in the set can be written as a linear combination of the others.

This is crucial in determining if vectors span a space, such as \(\mathbb{R}^3\). In our example, the vectors \( \mathbf{a} = (1, 3, 1) \), \( \mathbf{b} = (2, 1, 1) \), and \( \mathbf{c} = (2, -1, -5) \) are tested for linear independence by calculating the determinant, which is non-zero. Thus, they're linearly independent.

• Linear independence ensures that each vector adds a new dimension of information or directional component.
• If vectors are dependent, you can remove one without losing the ability to describe the space.