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Orthogonal vectors are vectors that are at right angles to each other in a given space. To determine if two vectors are orthogonal, we look at their dot product. If the dot product of two vectors is zero, they are orthogonal. This is because a zero dot product signifies that the vectors are perpendicular, sharing no direction in common.
For example, consider vectors \( \mathbf{u} = \begin{bmatrix} \frac{3}{5} \ \frac{4}{5} \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} -\frac{4}{5} \ \frac{3}{5} \end{bmatrix} \). Calculating the dot product, we have:

• \( \mathbf{u} \cdot \mathbf{v} = \frac{3}{5} \times \left(-\frac{4}{5}\right) + \frac{4}{5} \times \frac{3}{5} \) = \( -\frac{12}{25} + \frac{12}{25} = 0 \)

Since the result is zero, \( \mathbf{u} \) and \( \mathbf{v} \) are indeed orthogonal vectors.
This property is fundamental in mathematical computations, especially in simplifying problems involving multiple vectors.

Vector normalization is the process of adjusting the vector's length, or magnitude, so that it becomes 1, creating a unit vector. This is significant because unit vectors retain the direction of the original vector but adhere to a standardized length.
Great for usage in numerous fields like computer graphics and physics. The formula for normalizing a vector \( \mathbf{w} \) is:

• \( \mathbf{w}_{norm} = \frac{\mathbf{w}}{\|\mathbf{w}\|} \)

Where \( \|\mathbf{w}\| \) is the magnitude of the vector. For vectors \( \mathbf{u} \) and \( \mathbf{v} \) from our example:

• \( \| \mathbf{u} \| = \sqrt{ \left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 } = 1 \)
• \( \| \mathbf{v} \| = \sqrt{ \left(-\frac{4}{5}\right)^2 + \left(\frac{3}{5}\right)^2 } = 1 \)

Both magnitudes equal 1, so \( \mathbf{u} \) and \( \mathbf{v} \) are already unit vectors and thus normalized.

Calculating the dot product of two vectors helps in determining their relationship regarding direction. It's a fundamental operation in vector algebra and is used in computing angles between vectors, projections, and more.

The formula to compute the dot product for two vectors \( \mathbf{a} = \begin{bmatrix} a_1 \ a_2 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} b_1 \ b_2 \end{bmatrix} \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1 \times b_1 + a_2 \times b_2 \)

As shown in our problem, the vectors

• \( \mathbf{u} = \begin{bmatrix} \frac{3}{5} \ \frac{4}{5} \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} -\frac{4}{5} \ \frac{3}{5} \end{bmatrix} \)

produced a dot product of zero,
indicating orthogonal vectors. Understanding the dot product is crucial for analyzing vector sets and their dependencies or independencies.