Determine whether each pair of vectors is orthogonal.
√5i - 2j, -5i + 2 √5j

Verified step by step guidance

1. Identify the given vectors: \( \vec{a} = \sqrt{5}\mathbf{i} - 2\mathbf{j} \) and \( \vec{b} = -5\mathbf{i} + 2\sqrt{5}\mathbf{j} \).
2. Recall that two vectors are orthogonal if their dot product is zero.
3. Calculate the dot product of \( \vec{a} \) and \( \vec{b} \) using the formula: \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 \), where \( a_1 \) and \( a_2 \) are the components of \( \vec{a} \), and \( b_1 \) and \( b_2 \) are the components of \( \vec{b} \).
4. Substitute the components into the dot product formula: \( (\sqrt{5})(-5) + (-2)(2\sqrt{5}) \).
5. Simplify the expression to determine if the dot product equals zero, confirming orthogonality.

Key Concepts

Orthogonal Vectors

Two vectors are considered orthogonal if their dot product equals zero. This means that the angle between them is 90 degrees, indicating that they are perpendicular to each other in a geometric sense. Understanding this concept is crucial for determining the relationship between the given vectors.

Dot Product

The dot product of two vectors is calculated by multiplying their corresponding components and then summing those products. For vectors A = (a1, a2) and B = (b1, b2), the dot product is A · B = a1*b1 + a2*b2. This operation is fundamental in assessing whether two vectors are orthogonal.

Vector Representation

Vectors can be represented in component form, typically as a combination of unit vectors i (for the x-axis) and j (for the y-axis). In this case, the vectors √5i - 2j and -5i + 2√5j are expressed in terms of their i and j components, which is essential for performing operations like the dot product.