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

Verified step by step guidance

1. Recall that two vectors are orthogonal if their dot product equals zero.
2. Write the given vectors in component form: the first vector is \(\left(\sqrt{5}, -2\right)\) and the second vector is \(\left(-5, 2\sqrt{5}\right)\).
3. Calculate the dot product of the two vectors using the formula \(\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2\).
4. Substitute the components into the dot product formula: \(\left(\sqrt{5}\right) \times (-5) + (-2) \times \left(2\sqrt{5}\right)\).
5. Simplify the expression and check if the result equals zero; if it does, the vectors are orthogonal, otherwise they are not.

Key Concepts

Vector Components and Notation

Vectors in two dimensions are expressed using unit vectors i and j, representing the x and y directions respectively. Each vector can be written as a combination of these components, such as ai + bj, where a and b are scalar values indicating magnitude along each axis.

Dot Product of Vectors

The dot product is a scalar value calculated by multiplying corresponding components of two vectors and summing the results. For vectors a = a1i + a2j and b = b1i + b2j, the dot product is a1*b1 + a2*b2. It is used to determine the angle relationship between vectors.

Orthogonality of Vectors

Two vectors are orthogonal if their dot product equals zero, meaning they are perpendicular to each other. This property is fundamental in geometry and physics to identify right angles between directions or forces.