Determine whether each pair of vectors is orthogonal.
i + 3√2j, 6i - √2j

Verified step by step guidance

1. Step 1: Recall that two vectors are orthogonal if their dot product is zero.
2. Step 2: Identify the components of the vectors. The first vector is \( \mathbf{v_1} = \langle 1, 3\sqrt{2} \rangle \) and the second vector is \( \mathbf{v_2} = \langle 6, -\sqrt{2} \rangle \).
3. Step 3: Use the formula for the dot product of two vectors \( \mathbf{v_1} = \langle a_1, b_1 \rangle \) and \( \mathbf{v_2} = \langle a_2, b_2 \rangle \), which is \( a_1 \cdot a_2 + b_1 \cdot b_2 \).
4. Step 4: Substitute the components of the vectors into the dot product formula: \( 1 \cdot 6 + 3\sqrt{2} \cdot (-\sqrt{2}) \).
5. Step 5: Simplify the expression to determine if the dot product equals zero, which will confirm if the vectors are orthogonal.

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 = ai + bj and B = ci + dj, the dot product is given by A · B = ac + bd. This operation is fundamental in assessing whether two vectors are orthogonal.

Vector Components

Vectors can be expressed in terms of their components along the coordinate axes, typically represented as i (horizontal) and j (vertical) components. For example, the vector i + 3√2j has components of 1 and 3√2. Recognizing these components is essential for performing calculations like the dot product.