Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Let \(\mathbf{u}=\langle- 2,1\rangle, \mathbf{v}=\langle 3,4\rangle,\) and \(\mathbf{w}=\langle- 5,12\rangle .\) Evaluate each expression. $$\mathbf{u} \cdot(\mathbf{v}-\mathbf{w})$$

Step-by-Step Solution

Answer

\( \mathbf{u} \cdot(\mathbf{v}-\mathbf{w}) = -24 \).

Step 1: Understand the Dot Product

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \). We will use this in the final step.

Step 2: Subtract Vectors

First, calculate \( \mathbf{v} - \mathbf{w} \). Using vector subtraction, we subtract each corresponding component: \( \mathbf{v} - \mathbf{w} = \langle 3 - (-5), 4 - 12 \rangle = \langle 8, -8 \rangle \).

Step 3: Calculate the Dot Product

Now we calculate the dot product \( \mathbf{u} \cdot (\mathbf{v} - \mathbf{w}) \). Substitute \( \mathbf{u} = \langle -2, 1 \rangle \) and \( \mathbf{v} - \mathbf{w} = \langle 8, -8 \rangle \): \ \(-2 \cdot 8 + 1 \cdot (-8) = -16 - 8 = -24 \).

Key Concepts

Dot Product

The dot product is a way to multiply two vectors, resulting in a scalar. It is performed by multiplying corresponding components of the vectors and then summing these products. For vectors \[ \mathbf{a} = \langle a_1, a_2 \rangle \] and \[ \mathbf{b} = \langle b_1, b_2 \rangle, \] the dot product is calculated as \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2. \]
In the context of our previous exercise, we used vectors \( \mathbf{u} = \langle -2, 1 \rangle \) and \( \mathbf{v} - \mathbf{w} = \langle 8, -8 \rangle \), resulting in a dot product of \[-16 + (-8) = -24.\]

• Results in a scalar: The outcome is a single number, not a vector.
• Used in various applications: Physical phenomena like work, where force and distance are involved, utilize dot product.

Vector Subtraction

Vector subtraction is one of the foundational operations in vector algebra. In this process, we calculate the difference between vectors by subtracting their corresponding components. For vectors \( \mathbf{v} = \langle v_1, v_2 \rangle \) and \( \mathbf{w} = \langle w_1, w_2 \rangle \), their difference is computed as \[ \mathbf{v} - \mathbf{w} = \langle v_1 - w_1, v_2 - w_2 \rangle. \]
From the given exercise, \( \mathbf{v} = \langle 3, 4 \rangle \) and \( \mathbf{w} = \langle -5, 12 \rangle, \) the subtraction yields \( \mathbf{v} - \mathbf{w} = \langle 8, -8 \rangle. \)

• Important for vector operations: Subtraction finds application in calculating relative positions and changes in physics.
• Keeps vector format: The result of vector subtraction is another vector.

Vectors in Precalculus

Vectors play a crucial role in precalculus, bridging algebra and calculus by providing a visual and analytical tool. They are not just arrows in diagrams; they represent quantities with both magnitude and direction. This makes them suitable for various applications like physics, engineering, and computer graphics.
In precalculus, vectors are often used to:

• Represent physical quantities such as force, velocity, or displacement.
• Simplify complex problems into manageable equations through operations like addition, subtraction, and dot product.
• Analyze multi-dimensional systems by breaking them into component vectors.

Learning to manipulate vectors through operations like dot product and subtraction helps build the foundation for understanding more advanced concepts in calculus and beyond. They enhance spatial reasoning and problem-solving skills, essential for higher-level mathematics and various real-world applications.