Problem 26 - Algebra and Trigonometry

Question

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Find their dot product \(\mathbf{u} \cdot \mathbf{v} .\) $$ \mathbf{u}=\langle- 3,0,4\rangle, \quad \mathbf{v}=\left\langle 2,4, \frac{1}{2}\right\rangle $$

Step-by-Step Solution

Answer

The dot product \( \mathbf{u} \cdot \mathbf{v} \) is \(-4\).

Step 1: Understand the Dot Product

The dot product of two vectors \( \mathbf{u} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{v} = \langle b_1, b_2, b_3 \rangle \) is given by the formula \( \mathbf{u} \cdot \mathbf{v} = a_1b_1 + a_2b_2 + a_3b_3 \). We will apply this formula to find the dot product of \( \mathbf{u} \) and \( \mathbf{v} \).

Step 2: Identify the Components

Identify the components of the vectors: \( \mathbf{u} = \langle -3, 0, 4 \rangle \) implies \( a_1 = -3 \), \( a_2 = 0 \), \( a_3 = 4 \) and \( \mathbf{v} = \langle 2, 4, \frac{1}{2} \rangle \) implies \( b_1 = 2 \), \( b_2 = 4 \), \( b_3 = \frac{1}{2} \).

Step 3: Calculate Each Component Product

Compute the product of the corresponding components: \(-3 \times 2 = -6\), \(0 \times 4 = 0\), and \(4 \times \frac{1}{2} = 2\).

Step 4: Sum the Component Products

Add the results of the component products to find the dot product: \(-6 + 0 + 2 = -4\).

Key Concepts

Vectors

Vectors are fundamental objects in mathematics, physics, and engineering that represent quantities having both magnitude and direction. Unlike scalars, which only have magnitude, vectors provide more comprehensive information about an object's movement or position.

Vectors are usually depicted as directed line segments, with an arrow indicating direction and length indicating magnitude. In the Cartesian coordinate system, vectors are expressed in terms of their components. For instance, vector \( \mathbf{u} = \langle -3, 0, 4 \rangle \) means it has components \(-3\), \(0\), and \(4\) aligned with the \(x\), \(y\), and \(z\) axes, respectively. This helps in describing the vector's position in 3-dimensional space.

When we deal with vectors notations like \( \langle a, b, c \rangle \), we are often interested in operations like addition, subtraction, and calculating dot products. These operations are vital for problem-solving across various fields.

Vector Components

Understanding vector components is crucial for performing calculations or depicting physical phenomena like forces and velocities. Each vector is often broken down into its components along the axes of the coordinate system it resides in, such as \(x\), \(y\), and \(z\) in 3-dimensional space.

Let's consider vector \(\mathbf{u} = \langle -3, 0, 4 \rangle\):

• -3: This is the component along the x-axis. It tells us how far and in which direction the vector stretches horizontally.
• 0: The y-component indicates no movement along the y-axis.
• 4: The z-component shows the extent of movement along the vertical axis.

Knowing these components is pivotal when calculating the dot product or any vector operation, as they directly influence the result. Similarly, identifying the vector \( \mathbf{v} = \langle 2, 4, \frac{1}{2} \rangle \) provides an understanding of this vector’s behavior and interaction within 3D space.

Scalar Multiplication

Scalar multiplication is an operation where a vector is multiplied by a scalar (a single number). This alters the vector's magnitude without affecting its direction unless the scalar is negative, in which case the direction is reversed.

When computing a dot product, which involves scalar multiplication, it is essential to multiply each component of one vector by the corresponding component of another. For instance, the process transforms each vector into a product of scalars:\(-3 \times 2 = -6\), \(0 \times 4 = 0\), and \(4 \times \frac{1}{2} = 2\).

Finally, adding these scalar products, \(-6 + 0 + 2\), results in the dot product of \(-4\). This dot product represents a measure of similarity between the two vectors in terms of their direction and size. It has many applications, such as determining whether two vectors are perpendicular or finding the angle between them. Understanding scalar multiplication is key to mastering vector mathematics.