In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.v = i + j, w = i - j

Verified step by step guidance

1. Step 1: Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if their dot product is zero.
2. Step 2: Write the vectors \( \mathbf{v} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} - \mathbf{j} \) in component form: \( \mathbf{v} = (1, 1) \) and \( \mathbf{w} = (1, -1) \).
3. Step 3: Use the formula for the dot product of two vectors \( \mathbf{v} = (a_1, b_1) \) and \( \mathbf{w} = (a_2, b_2) \), which is \( \mathbf{v} \cdot \mathbf{w} = a_1a_2 + b_1b_2 \).
4. Step 4: Substitute the components of \( \mathbf{v} \) and \( \mathbf{w} \) into the dot product formula: \( (1)(1) + (1)(-1) \).
5. Step 5: Simplify the expression to determine if the dot product is zero, which will confirm if the vectors are orthogonal.

Key Concepts

Dot Product

The dot product is a mathematical operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results. For vectors v = (v1, v2) and w = (w1, w2), the dot product is given by v · w = v1*w1 + v2*w2. This operation is crucial for determining the angle between vectors and checking for orthogonality.

Orthogonal Vectors

Two vectors are considered orthogonal if they are perpendicular to each other, which occurs when their dot product equals zero. This property is significant in various applications, including physics and computer graphics, as it indicates that the vectors do not influence each other in their respective directions. Understanding orthogonality helps in simplifying problems involving vector components.

Vector Representation

Vectors can be represented in terms of their components along the coordinate axes. In this case, the vectors v = i + j and w = i - j can be expressed as v = (1, 1) and w = (1, -1). This representation allows for straightforward calculations, such as the dot product, and aids in visualizing the vectors in a Cartesian plane, facilitating the analysis of their relationships.