In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.v = 12i - 8j, w = 2i + 3j

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 formula for the dot product of two vectors \( \mathbf{v} = a_1\mathbf{i} + b_1\mathbf{j} \) and \( \mathbf{w} = a_2\mathbf{i} + b_2\mathbf{j} \): \( \mathbf{v} \cdot \mathbf{w} = a_1a_2 + b_1b_2 \).
3. Step 3: Substitute the given vectors \( \mathbf{v} = 12\mathbf{i} - 8\mathbf{j} \) and \( \mathbf{w} = 2\mathbf{i} + 3\mathbf{j} \) into the dot product formula.
4. Step 4: Calculate the dot product: \( 12 \times 2 + (-8) \times 3 \).
5. Step 5: Determine if the result from Step 4 is zero. If it is, then \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and returns a single scalar value. 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 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. In the context of the given vectors, checking for orthogonality involves calculating their dot product and verifying if it equals zero.

Vector Components

Vectors in a two-dimensional space can be expressed in terms of their components along the x-axis and y-axis, typically represented as v = ai + bj. Here, 'a' and 'b' are the scalar components that indicate the vector's magnitude in the respective directions. Understanding vector components is essential for performing operations like the dot product, as it allows for the systematic calculation of the interaction between vectors in a Cartesian coordinate system.