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Chapter 6: Problem 10

In Exercises \(7-14,\) find \(\mathbf{u} \cdot \mathbf{v}\) $$\mathbf{u}=\langle- 2,5\rangle$$ $$\mathbf{v}=\langle- 1,-8\rangle$$

Short Answer

Expert verified
The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is -38.

Step by step solution

01

Identify the components of the vectors

The components of vector \(\mathbf{u}\) are -2 and 5, and the components of vector \(\mathbf{v}\) are -1 and -8.
02

Multiply the corresponding components

Now, multiply the corresponding components of the vectors: (-2) * (-1) and 5 * (-8). This will give 2 and -40 respectively.
03

Add the products

Now add the products obtained in step 2. Hence, \( 2 + (-40) = -38 \)

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Key Concepts

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

Understanding Vector Components
Vectors are essential in mathematics and physics. They help represent direction and magnitude. A vector in two dimensions has two components: an x-component and a y-component. Each component represents a part of the vector along the x-axis and y-axis, respectively.
In our example, vector \( \mathbf{u} = \langle -2, 5 \rangle \) has components -2 and 5.
This means:
  • The x-component of \( \mathbf{u} \) is -2, showing movement to the left.
  • The y-component of \( \mathbf{u} \) is 5, indicating movement upwards.
For vector \( \mathbf{v} = \langle -1, -8 \rangle \), the components -1 and -8 mean it moves left and down.
Understanding these components is crucial for performing operations like the dot product."},{
Performing Vector Operations: Dot Product
Vector operations are mathematical procedures performed on vectors.
The dot product is a common operation. It combines two vectors into a single scalar value. This result gives an indication of how aligned the vectors are. For two vectors \( \mathbf{u} = \langle a_1, b_1 \rangle \) and \( \mathbf{v} = \langle a_2, b_2 \rangle \), the dot product is calculated as:
  • Multiply the x-components: \( a_1 \times a_2 \)
  • Multiply the y-components: \( b_1 \times b_2 \)
  • Add the results: \( (a_1 \times a_2) + (b_1 \times b_2) \)
In the example, \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle -1, -8 \rangle \):
  • \((-2) \times (-1) = 2\)
  • \(5 \times (-8) = -40\)
Adding them gives \(2 + (-40) = -38\).
This dot product shows how vectors relate in terms of direction and length.
Mathematical Calculations for Dot Products
Calculating the dot product involves simple arithmetic, yet it's a powerful tool. Here's how it's done step-by-step using vectors \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle -1, -8 \rangle \):
Firstly, identify the vector components. Here, \( \mathbf{u} \) has components -2 and 5; \( \mathbf{v} \) has -1 and -8.
Next, multiply corresponding components:
  • \((-2) \cdot (-1) = 2\)
  • \(5 \cdot (-8) = -40\)
Finally, add these products together:
  • \(2 + (-40) = -38\)
This final result, -38, is the scalar "dot product" of the vectors.
This operation can help you determine angles and projections between vectors.

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Most popular questions from this chapter

Vector Operations In Exercises \(57-62\) , find the component form of \(v\) and sketch the specified vector operations geometrically, where \(u=2 i-j\) and \(w=i+2 j\) $$\mathbf{v}=\frac{1}{2}(3 \mathbf{u}+\mathbf{w})$$

True or False?, determine whether the statement is true or false. Justify your answer. Writing In Exercise \(64,\) the Law of Cosines was used to solve a triangle in the two-solution case of SSA. Can the Law of Cosines be used to solve the no-solution and single-solution cases of SSA? Explain.

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