In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.v = 2i - 2j, w = -i + j

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1. Calculate the dot product of \( \mathbf{v} \) and \( \mathbf{w} \) using the formula: \( \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 \).
2. Substitute the components of \( \mathbf{v} = 2\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{w} = -\mathbf{i} + \mathbf{j} \) into the dot product formula.
3. Perform the multiplication for each component: \( 2 \times (-1) \) and \( -2 \times 1 \).
4. Add the results of the multiplications to find the dot product.
5. Determine if the dot product is zero. If it is, then \( \mathbf{v} \) and \( \mathbf{w} \) 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 = ai + bj and w = ci + dj, the dot product is given by v · w = ac + bd. 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 engineering, 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 Components

Vectors can be expressed in terms of their components along the coordinate axes, typically represented as i (x-axis) and j (y-axis) in two dimensions. For example, the vector v = 2i - 2j has components 2 and -2. Analyzing these components is essential for performing operations like the dot product and understanding the geometric interpretation of vectors in space.