In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.v = 3i, w = -4i

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1. Identify the vectors \( \mathbf{v} = 3\mathbf{i} \) and \( \mathbf{w} = -4\mathbf{i} \).
2. Recall that two vectors are orthogonal if their dot product is zero.
3. The dot product of two vectors \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{w} = c\mathbf{i} + d\mathbf{j} \) is given by \( \mathbf{v} \cdot \mathbf{w} = ac + bd \).
4. Substitute the components of \( \mathbf{v} \) and \( \mathbf{w} \) into the dot product formula: \( 3 \times (-4) + 0 \times 0 \).
5. Evaluate the expression to determine if the dot product is zero, which would indicate that 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 the 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 their dot product equals zero. This means that they are at right angles (90 degrees) to each other in a geometric sense. In practical terms, if v · w = 0, then the vectors do not influence each other in their respective directions, which is a key concept in various applications, including physics and engineering.

Vector Representation

Vectors can be represented in a coordinate system using unit vectors, such as i, j, and k for three-dimensional space. In this case, v = 3i and w = -4i are both one-dimensional vectors along the x-axis. Understanding vector representation is essential for performing operations like the dot product and visualizing the relationship between vectors in space.