In the context of geometric vectors, which of the following best describes orthogonal lines?

• A. They are lines whose direction vectors have a dot product of $0$.
• B. They are lines that are parallel to each other.
• C. They are lines whose direction vectors have a dot product of $1$.
• D. They are lines that intersect at an angle of $45°$.

Answer: (A)

Verified step by step guidance

1. Recall that two lines are orthogonal if they intersect at a right angle, which means the angle between them is 90 degrees.
2. Understand that the direction vectors of lines represent their orientation in space, and the angle between two vectors can be found using their dot product.
3. Use the dot product formula for two vectors \( \mathbf{a} \) and \( \mathbf{b} \): \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \] where \( \theta \) is the angle between the vectors.
4. Since orthogonal lines meet at 90 degrees, substitute \( \theta = 90^\circ \) into the formula, noting that \( \cos(90^\circ) = 0 \). This implies: \[ \mathbf{a} \cdot \mathbf{b} = 0 \]
5. Conclude that the best description of orthogonal lines is that their direction vectors have a dot product of zero.