Problem 8 - APEX Calculus

Question

Find the dot product of the given vectors. \(\vec{u}=\langle 3,5,-1\rangle, \vec{v}=\langle 4,-1,7\rangle\)

Step-by-Step Solution

Answer

The dot product is 0.

Step 1: Write Down the Dot Product Formula

The dot product of two vectors \( \vec{u} \) and \( \vec{v} \) is calculated using the formula:\[\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3.\]Here, the vectors \( \vec{u} = \langle 3, 5, -1 \rangle \) and \( \vec{v} = \langle 4, -1, 7 \rangle \). Substitute the components into the formula to find the dot product.

Step 2: Substitute Vector Components

Substitute the components of vectors \( \vec{u} \) and \( \vec{v} \) into the dot product formula:\[\vec{u} \cdot \vec{v} = (3)(4) + (5)(-1) + (-1)(7).\] This expression needs to be simplified to calculate the dot product.

Step 3: Perform Multiplication

Calculate the products of the corresponding components:1. \( 3 \times 4 = 12 \).2. \( 5 \times (-1) = -5 \).3. \( -1 \times 7 = -7 \).Substitute these results back into the expression.

Step 4: Simplify the Expression

Add the results from the multiplication to get the final dot product:\[12 + (-5) + (-7) = 12 - 5 - 7 = 0.\]Thus, the dot product of \( \vec{u} \) and \( \vec{v} \) is 0.

Key Concepts

Vectors

Vectors are fundamental mathematical objects used to describe quantities that have both magnitude and direction. Imagine arrows in a three-dimensional space; these arrows represent vectors. Each vector has components along each axis of the space, usually denoted with labels like \( x \), \( y \), and \( z \) for three dimensions.

In essence, a vector in 3D space is written as \( \langle x, y, z \rangle \), where:\

• \( x \) is the component along the X-axis
• \( y \) is the component along the Y-axis
• \( z \) is the component along the Z-axis

Vectors are particularly useful for representing physical quantities such as velocity, force, and displacement in physics, among other fields in sciences and engineering.

Vector Multiplication

In mathematics, particularly in linear algebra, vector multiplication is a fundamental operation that combines two vectors. There are two popular types of vector multiplication: the dot product and the cross product.

• Dot Product: Also known as the scalar product, it multiplies two vectors to result in a scalar. This is what is used in this particular exercise.
• Cross Product: Unlike the dot product, this results in another vector that is perpendicular to the two original vectors. However, it is only defined in three-dimensional space.

Understanding these operations is vital because they are used in various calculations in physics, computer graphics, engineering, and many other applications.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and linear transformations. It forms the basis for understanding more complex mathematical concepts.

Here are some key components of linear algebra:

• Vector Spaces: Collections of vectors that can be added together and multiplied by scalars.
• Matrices: Rectangular arrays of numbers that can represent linear transformations and systems of linear equations.
• Functions: Used to map aspects of a vector space to another.

Linear algebra is used extensively in computer science, engineering, and statistics, making it an essential subject for students in various fields.

Dot Product Calculation

The dot product is a specific type of vector multiplication that produces a single number (scalar). It provides valuable information about the angle between two vectors: if the dot product is zero, the vectors are orthogonal.

The formula for the dot product of two vectors \( \vec{u} = \langle u_1, u_2, u_3 \rangle \) and \( \vec{v} = \langle v_1, v_2, v_3 \rangle \) is:
\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]

For the vectors \( \vec{u} = \langle 3, 5, -1 \rangle \) and \( \vec{v} = \langle 4, -1, 7 \rangle \), we perform these calculations:

• Find the product of the corresponding components: \( (3)(4), (5)(-1), (-1)(7) \)
• Add these products: \( 12, -5, -7 \)
• Sum them up to get the final dot product: \( 12 - 5 - 7 = 0 \)

The result being zero indicates that vector \( \vec{u} \) and vector \( \vec{v} \) are perpendicular to each other.