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In the context of vectors, 'parallel' means that one vector is a scaled version of the other. This implies that they point in the same or exactly opposite directions. To determine if vectors \( \textbf{u} = \langle 2, 3 \rangle \) and \( \textbf{v} = \langle 8, 12 \rangle \) are parallel, we check for a scalar multiple. If we find a scalar \( k \) such that \( \textbf{v} = k \textbf{u} \), the vectors are parallel.
By verifying the components, we see that \( 8 = 4 \times 2 \) and \( 12 = 4 \times 3 \). Here, \( k = 4 \). Hence, \( \textbf{v} \) is obtained by scaling \( \textbf{u} \) by 4. They are parallel because of this consistent scaling factor. Parallel vectors indicate alignment in the same direction or the exact opposite, depending on the sign of the scalar.

Vectors are considered perpendicular (orthogonal) if the angle between them is 90 degrees. A key characteristic of perpendicular vectors is that their dot product equals zero. The dot product of two vectors \( \textbf{u} = \langle u_1, u_2 \rangle \) and \( \textbf{v} = \langle v_1, v_2 \rangle \) is calculated as: \[ \textbf{u} \cdot \textbf{v} = u_1 v_1 + u_2 v_2 \] Checking for the vectors \( \textbf{u} = \langle 2, 3 \rangle \) and \( \textbf{v} = \langle 8, 12 \rangle \), we find: \[ \textbf{u} \cdot \textbf{v} = 2 \times 8 + 3 \times 12 = 16 + 36 = 52 \] Since the dot product is \ not zero \, the vectors are not perpendicular. The zero value of the dot product signifies a perfect 90-degree angle between the vectors, which is not the case here.

The dot product (or scalar product) of two vectors quantifies the extent to which they point in the same direction. It can be calculated using the formula: \[ \textbf{u} \cdot \textbf{v} = u_1 v_1 + u_2 v_2 \] For vectors \( \textbf{u} = \langle 2, 3 \rangle \) and \( \textbf{v} = \langle 8, 12 \rangle \), the dot product is: \[ 2 \times 8 + 3 \times 12 = 52 \] The dot product helps determine various properties. If its value is zero, the vectors are perpendicular. If non-zero, it provides a measure of how closely the vectors align directionally. Higher values indicate greater alignment. In cases like these, understanding the dot product is essential for analyzing the geometric relationships between vectors.