Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{0} \cdot \mathbf{u}=0\)

Step-by-Step Solution

Answer

The dot product \( \mathbf{0} \cdot \mathbf{u} = 0 \).

Step 1: Understand the Property

We need to prove that the dot product of the zero vector \( \mathbf{0} \) and any vector \( \mathbf{u} = \langle u_1, u_2 \rangle \) is zero. The zero vector in two dimensions is \( \mathbf{0} = \langle 0, 0 \rangle \).

Step 2: Define 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 defined as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

Step 3: Apply the Dot Product Formula

Using the definition, substitute \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{0} = \langle 0, 0 \rangle \) into the dot product formula: \( \mathbf{0} \cdot \mathbf{u} = 0 \cdot u_1 + 0 \cdot u_2 \).

Step 4: Simplify the Expression

Compute the expression from Step 3: \( 0 \cdot u_1 + 0 \cdot u_2 = 0 + 0 = 0 \).

Step 5: Conclusion

We have shown that \( \mathbf{0} \cdot \mathbf{u} = 0 \). The dot product of a zero vector and any other vector is always zero.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics that combines two vectors to produce a scalar. For any two-dimensional vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), the dot product is calculated by multiplying their corresponding components and summing the results, specifically: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \] This operation tells us about the relationship between the vectors; for instance:

• If the dot product is zero, the vectors are orthogonal, or at right angles to each other.
• A positive dot product indicates the vectors have a component pointing in the same direction.
• A negative dot product means they point in opposite directions.

Understanding the dot product is crucial for exploring vector properties and operations.

Zero Vector

The zero vector, denoted \(\mathbf{0}\), is a unique vector whose magnitude and direction are both nonexistent. It is represented in two dimensions as \(\mathbf{0} = \langle 0, 0 \rangle\). One of its significant characteristics is how it interacts with other vectors through operations like the dot product.
When you take the dot product of the zero vector with any other vector \(\mathbf{u} = \langle u_1, u_2 \rangle\), according to the step by step solution, the result is always zero:

• The computation follows this logic: \( \mathbf{0} \cdot \mathbf{u} = 0 \cdot u_1 + 0 \cdot u_2 = 0 + 0 = 0 \).
• This is because multiplying any number by zero results in zero.

The concept of the zero vector is pivotal in linear algebra and vector spaces, serving as the additive identity.

Vector Operations

Vector operations are mathematical procedures you can perform on vectors, such as addition, subtraction, and the dot product. These operations help us manipulate vectors in ways that are useful in physics, engineering, and computer graphics.

• Addition: Combining vectors by adding their corresponding components. For \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\), their sum is \(\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle\).
• Subtraction: Subtracting one vector from another involves subtracting each component. \(\mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2 \rangle\).
• Dot Product: As previously discussed, this combines two vectors into a single scalar value and depends on the angle between them.

These operations are not just theoretical constructs; they form the backbone of various applications, from calculating forces in physics to transforming objects in computer graphics.