Determine whether each pair of vectors is orthogonal.
〈1, 1〉, 〈1, -1〉

Verified step by step guidance

1. Calculate the dot product of the two vectors.
2. The dot product of vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is given by \( a \cdot c + b \cdot d \).
3. Substitute the components of the vectors \( \langle 1, 1 \rangle \) and \( \langle 1, -1 \rangle \) into the dot product formula.
4. Perform the multiplication and addition: \( 1 \cdot 1 + 1 \cdot (-1) \).
5. If the dot product is zero, the vectors are orthogonal.

Key Concepts

Orthogonal Vectors

Two vectors are considered orthogonal if their dot product equals zero. This means that they are at right angles to each other in a geometric sense. In a two-dimensional space, if vector A is represented as 〈a1, a2〉 and vector B as 〈b1, b2〉, the dot product is calculated as a1*b1 + a2*b2.

Dot Product

The dot product of two vectors is a scalar value obtained by multiplying their corresponding components and summing the results. For vectors 〈a1, a2〉 and 〈b1, b2〉, the dot product is calculated as a1*b1 + a2*b2. This operation is fundamental in determining the angle between vectors and checking for orthogonality.

Vector Representation

Vectors can be represented in coordinate form, such as 〈x, y〉 in two dimensions. Each component corresponds to a position along the respective axes. Understanding how to interpret and manipulate these components is essential for performing operations like the dot product and determining relationships between vectors, such as orthogonality.