Problem 22

Question

Find the indicated quantity, assuming $\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},$ and $\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}$. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})$$

Step-by-Step Solution

Verified

Answer

The dot product is 9.

1Step 1: Understand Vector Addition

First, we need to determine the vector sum $ \mathbf{v} + \mathbf{w} $. Given $ \mathbf{v} = \mathbf{i} - 3 \mathbf{j} $ and $ \mathbf{w} = 3 \mathbf{i} + 4 \mathbf{j} $, we add the corresponding components.

2Step 2: Compute Vector Sum

Add the $ \mathbf{i} $ components: $ 1 + 3 = 4 $. Add the $ \mathbf{j} $ components: $ -3 + 4 = 1 $. Thus, $ \mathbf{v} + \mathbf{w} = 4 \mathbf{i} + \mathbf{j} $.

3Step 3: Set up the Dot Product

The next step is to find the dot product of $ \mathbf{u} $ and $ \mathbf{v} + \mathbf{w} $. $ \mathbf{u} = 2 \mathbf{i} + \mathbf{j} $ and $ \mathbf{v} + \mathbf{w} = 4 \mathbf{i} + \mathbf{j} $.

4Step 4: Calculate the Dot Product

Use the dot product formula: $ (a \mathbf{i} + b \mathbf{j}) \cdot (c \mathbf{i} + d \mathbf{j}) = ac + bd $. Here, $ a = 2, b = 1, c = 4, d = 1 $. Calculate: $ (2 \times 4) + (1 \times 1) = 8 + 1 = 9 $.

5Step 5: Verify the Result

Ensure all calculations are correct and align with the properties of vector operations. The computed dot product $ 9 $ reflects the operation $ \mathbf{u} \cdot(\mathbf{v} + \mathbf{w}) $.

Key Concepts

Vector AdditionDot ProductVector Operations

Vector Addition

Vector addition involves combining two or more vectors to form a new vector. When vectors are expressed in terms of their components, like $ \mathbf{v} \, \text{and} \, \mathbf{w} $, we add corresponding components. For example, consider the vectors:

$ \mathbf{v} = \mathbf{i} - 3\mathbf{j} $
$ \mathbf{w} = 3\mathbf{i} + 4\mathbf{j} $

To find $ \mathbf{v} + \mathbf{w} $, simply add the $ \mathbf{i} $ components and the $ \mathbf{j} $ components separately:

$ 1 \,+\, 3, \, \text{for the} \, \mathbf{i} $ component, results in $ 4\mathbf{i} $
$-3 \,+\, 4, \, \text{for the} \, \mathbf{j} $ component, results in $ \mathbf{j} $

Combining results gives the summed vector: $ \mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j} $. This process highlights that vectors can be easily manipulated, helping to visualize problems in physics and engineering.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation that combines two vectors into a single scalar. It measures how much one vector extends in the direction of another. For two-dimensional vectors given as $ \mathbf{a} = a\mathbf{i} + b\mathbf{j} $ and $ \mathbf{c} = c\mathbf{i} + d\mathbf{j} $, the dot product formula is:\[ \mathbf{a} \cdot \mathbf{c} = ac + bd \]Let's apply this to vectors $ \mathbf{u} = 2\mathbf{i} + \mathbf{j} $ and $ \mathbf{v} + \mathbf{w} = 4\mathbf{i} + \mathbf{j} $:

Identify components: $ a = 2, \, b = 1, \, c = 4, \, d = 1 $
Compute: $ (2 \times 4) + (1 \times 1) = 8 + 1 = 9 $

The result is a scalar $ 9 $, indicating the magnitude of the projection of one vector onto another, and giving insights into the relation between their directions. The dot product is highly valuable in physics for calculating work and in computer graphics for shading effects.

Vector Operations

Vector operations consist of fundamental actions we perform on vectors, such as addition, subtraction, and dot product. These operations allow us to solve complex problems involving direction and magnitude in many fields.

Basic Vector Operations

Addition: Combine vectors by adding corresponding components, as shown previously with $ \mathbf{v} + \mathbf{w} $.
Subtraction: Similar to addition, but subtract the components of one vector from another.
Scalar Multiplication: Multiply each component of a vector by a scalar (a single number).

Applications

Physics: Analyzing forces, velocity, and acceleration.
Engineering: Designing structural and mechanical systems.
Computer Graphics: Rendering images, modeling movement and rotations.

Understanding these operations will empower you to handle various complex problems efficiently, employing vectors as tools to describe and model the physical world.

Problem 22

Other exercises in this chapter

Problem 22

Find an equation of the plane that passes through the points $P, Q,$ and $R$ $$P(3,4,5), \quad Q(1,2,3), \quad R(4,7,6)$$

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Problem 22

Express the given vector in terms of the unit vectors i, $\mathbf{j}$. and $\mathbf{k}$. $$\left\langle- a, \frac{1}{3} a, 4\right\rangle$$

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Problem 22

Sketch the given vector with initial point (4, 3), and find the terminal point. $$\mathbf{u}=\langle- 8,-1\rangle$$

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Problem 23

Find the area of the parallelogram determined by the given vectors. $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}+4 \mathbf{k}, \quad \mathbf{v}=\frac{1}{2} \mathbf{i}+2

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