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The cross product is an operation on two vectors in three-dimensional space, often symbolized by "\( \times \)". It results in a vector that is perpendicular to both original vectors, offering new ways of solving geometrical problems like finding the area of a parallelogram. The cross product is defined for two vectors \( \mathbf{a} \) and \( \mathbf{b} \) as:

• \( \mathbf{a} \times \mathbf{b} = \|\mathbf{a} \| \|\mathbf{b} \| \sin \theta \mathbf{n} \)

Here, \( \| \mathbf{a} \| \) and \( \| \mathbf{b} \| \) are magnitudes of vectors, \( \theta \) is the angle between them, and \( \mathbf{n} \) is a unit vector perpendicular to the plane formed by the original vectors.
To compute the cross product using determinants, we arrange \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) with the components of the vectors under them, forming a \( 3 \times 3 \) matrix. Its determinant gives the resulting vector:
From this, you can calculate each component by smaller determinants, eventually obtaining the resultant vectors' values.

The magnitude of a vector informs us of the vector's length or size, irrespective of its direction. It's a fundamental concept when working with vectors and is often expressed using the formula:

• For a vector \( \mathbf{v} = ai + bj + ck \), the magnitude is \( \| \mathbf{v} \| = \sqrt{a^2 + b^2 + c^2} \).

In our exercise, the cross product yields the vector \( 5\mathbf{i} - 14\mathbf{j} - 6\mathbf{k} \).
To determine the magnitude, compute as follows:

• Square each component: \( 5^2 = 25 \), \((-14)^2 = 196\), and \((-6)^2 = 36 \).
• Sum the squares: \( 25 + 196 + 36 = 257 \).
• Take the square root of the sum: \( \sqrt{257} \).

This value \( \sqrt{257} \) represents the size of the vector obtained from the cross product.
This magnitude is crucial as it leads directly to our next concept: calculating the area.

The area of a parallelogram formed by two vectors positioned to share a common point or base is calculated using their cross product, specifically the vector's magnitude. The rationale stems from the geometric definition of the cross product: it quantifies the area of the parallelogram spanned by the given vectors. In essence:

• Given vectors \( \mathbf{a} \) and \( \mathbf{b} \), their parallelogram's area is \( \| \mathbf{a} \times \mathbf{b} \| \). This is equivalent to the magnitude of their cross product vector.

For the vectors \( \mathbf{a} = -\mathbf{i} + \mathbf{j} - 3\mathbf{k} \) and \( \mathbf{b} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k} \), the area is determined by:

• Calculating \( \| 5\mathbf{i} - 14\mathbf{j} - 6\mathbf{k} \| = \sqrt{257} \).

This magnitude represents the area bounded by vectors \( \mathbf{a} \) and \( \mathbf{b} \), providing a clear connection between algebraic calculation and geometric interpretation.