Problem 39 - Contemporary Precalculus

Question

Let \(\boldsymbol{u}=\langle a, b\rangle, \boldsymbol{v}=\langle c, d\rangle,\) and \(\boldsymbol{w}=\langle r, s\rangle\) Verify that the given property of dot products is valid by calculating the quantities on each side of the equal sign. $$0 \cdot \mathbf{u}=0$$

Step-by-Step Solution

Answer

To verify the given property of dot products, we followed these steps: 1. Wrote the zero vector: \(\mathbf{0} = \langle 0, 0\rangle\). 2. Calculated the dot product of the zero vector and given vector \(\mathbf{u}\): \((\mathbf{0} \cdot \mathbf{u}) = \langle 0, 0\rangle \cdot \langle a, b\rangle = (0 \times a) + (0 \times b)\). 3. Simplified the dot product: \((\mathbf{0} \cdot \mathbf{u}) = 0 + 0 = 0\). 4. Verified the given property: \(0 \cdot \mathbf{u} = 0\). This calculation confirms that the dot product of the zero vector and any vector \(\mathbf{u}\) is always 0.

Step 1: Write the zero vector

The zero vector is a vector with all components as zero. In this case, since \(\mathbf{u}\) is a 2-dimensional vector, the zero vector is \(\mathbf{0} = \langle 0, 0\rangle\).

Step 2: Find the dot product 0 ⋅ 𝐮

The dot product of two vectors can be found by multiplying their corresponding components and adding the products. The dot product of the zero vector and vector \(\mathbf{u}\) is: $$(\mathbf{0} \cdot \mathbf{u}) = \langle 0, 0\rangle \cdot \langle a, b\rangle = (0 \times a) + (0 \times b)$$

Step 3: Simplify the dot product

As we can see in the dot product equation obtained in Step 2, the result is $$(\mathbf{0} \cdot \mathbf{u}) = (0 \times a) + (0 \times b) = 0 + 0 = 0$$

Step 4: Verify the property

Now we have calculated the dot product of the zero vector and vector \(\mathbf{u}\), which is 0. This confirms the given property of dot products, as \(0 \cdot \mathbf{u} = 0\).

Key Concepts

Vector Operations

Vector operations are fundamental in mathematics, particularly in algebra and geometry. They involve performing calculations with vectors, which are quantities that have both a direction and a magnitude. In two or three dimensions, vectors are often represented as an ordered list of numbers, like \(\langle a, b \rangle\) or \(\langle a, b, c \rangle\). There are several basic vector operations:

• Addition: To add vectors, simply add their corresponding components. For example, \(\mathbf{u} + \mathbf{v} = \langle a+c, b+d \rangle\) for vectors \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\).


• Subtraction: Similar to addition, but you subtract the components. For instance, \(\mathbf{u} - \mathbf{v} = \langle a-c, b-d \rangle\).


• Scalar Multiplication: This involves multiplying each component of a vector by a scalar value \(k\). So, \(k \cdot \mathbf{u} = \langle k \cdot a, k \cdot b \rangle\).


• Dot Product: This operation involves multiplying corresponding components of two vectors and summing the results, which is central to our exercise today. Mathematically, for vectors \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\), the dot product is calculated as \(\mathbf{u} \cdot \mathbf{v} = a \cdot c + b \cdot d\).

These basic operations are crucial in applying vectors to solve real-world problems in physics, engineering, and computer graphics.

Zero Vector

The zero vector, sometimes called the null vector, is special in the realm of vectors. It is defined as a vector with all of its components equal to zero. For example, the zero vector in two-dimensional space is \(\mathbf{0} = \langle 0, 0 \rangle\), while in three-dimensional space, it is \(\mathbf{0} = \langle 0, 0, 0 \rangle\).

• The zero vector is significant because it acts as the additive identity in vector operations. When any vector is added to the zero vector, the result is the original vector. Mathematically, \(\mathbf{u} + \mathbf{0} = \mathbf{u}\).


• It also plays a crucial role in multiplication, especially concerning the dot product. As seen in our exercise, multiplying any vector by the zero vector always results in a zero scalar, i.e., \(\mathbf{0} \cdot \mathbf{u} = 0\). This is because multiplying zero by any number yields zero, and the addition of these zeros is still zero.

This property makes the zero vector an essential concept, serving as a basis for more advanced vector algebra topics.

Properties of Dot Product

The dot product, also known as the scalar product, has several properties that make it vital in vector analysis. Understanding these properties enhances your ability to work with vectors in various theoretical and practical applications.

• Commutative Property: The dot product of two vectors is commutative, meaning \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\).


• Distributive Property: It satisfies the distributive property over vector addition: \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\).


• Scalar Multiplication: The dot product can be scaled by a scalar, such that \(k(\mathbf{u} \cdot \mathbf{v}) = (k\mathbf{u}) \cdot \mathbf{v}\).


• Angle Relation: The dot product relates to the cosine of the angle \(\theta\) between the two vectors: \(\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta)\), where \(||\mathbf{u}||\) and \(||\mathbf{v}||\) are the magnitudes of the vectors.

These properties not only assist in simplifying calculations but also provide insights into the geometric and physical interpretations of vectors in various disciplines.