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Chapter 8: Problem 73

Find the dot product for each pair of vectors. $$\langle 5,2\rangle,\langle- 4,10\rangle$$

Short Answer

Expert verified
The dot product is 0.

Step by step solution

01

- Recall the formula for the dot product

The dot product of two vectors \(\textbf{a} = \langle a_1, a_2\rangle\) and \(\textbf{b} = \langle b_1, b_2\rangle\) is given by \[ \textbf{a} \cdot \textbf{b} = a_1b_1 + a_2b_2 \]
02

- Identify components of the vectors

For \(\textbf{a} = \langle 5, 2\rangle\), the components are \(a_1 = 5\) and \(a_2 = 2\). For \(\textbf{b} = \langle -4, 10\rangle\), the components are \(b_1 = -4\) and \(b_2 = 10\).
03

- Multiply corresponding components

Calculate the products of the corresponding components: \(a_1b_1 = 5 \cdot (-4) = -20\) and \(a_2b_2 = 2 \cdot 10 = 20\).
04

- Sum the products

Add the products obtained in the previous step: \[ -20 + 20 = 0 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
Understanding vector components is the first step in solving problems involving vectors, like finding the dot product. A vector is often represented in terms of its components. These components are essentially just the parts of the vector that point in the directions of the coordinate axes.
For a 2-dimensional vector \(\textbf{a} = \langle a_1, a_2 \rangle\), the components are \(a_1\) and \(a_2\). In our exercise, the vectors \(\langle 5, 2 \rangle\) and \(\langle -4, 10 \rangle\) have \(a_1 = 5, a_2 = 2, b_1 = -4, \text{and} b_2 = 10\). Identifying these components correctly is crucial because they will be used in further calculations.
Vector Multiplication
When we talk about vector multiplication in the context of dot products, we mean multiplying the corresponding components of two vectors. This operation is different from the normal arithmetic multiplication because it involves each matching pair of components from the two vectors.
In our example, we multiply \(a_1 \text{with} b_1\) and \(a_2 \text{with} b_2\). Given the component pairs from the vectors \(\langle 5, 2 \rangle\) and \(\langle -4, 10 \rangle\), perform:
  • \(a_1b_1 = 5 \cdot (-4) = -20 \)
  • \(a_2b_2 = 2 \cdot 10 = 20 \)
This process is straightforward but requires attention to sign and value to avoid mistakes.
Sum of Products
The final step in finding the dot product is summing the products calculated from the corresponding components of the two vectors. This sum gives us the dot product. In our example:
We already calculated \(a_1b_1 = -20 \text{ and } a_2b_2 = 20\).
Adding these together, we get: \[-20 + 20 = 0\] Thus, the dot product for the vectors \(\langle 5, 2 \rangle\) and \(\langle -4, 10 \rangle\) is \(0\).
Summing the products is simple but it seals the deal in determining the dot product, which has various useful applications in physics and engineering.

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