Problem 2 - Advanced Engineering Mathematics

Question

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{b} \cdot \mathbf{c}\)

Step-by-Step Solution

Answer

\( \mathbf{b} \cdot \mathbf{c} = 4 \)

Step 1: Recall Dot Product Formula

The dot product of two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \) is calculated as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \). We will use this formula to find \( \mathbf{b} \cdot \mathbf{c} \).

Step 2: Identify Components of Vectors

Identify the components of vectors \( \mathbf{b} \) and \( \mathbf{c} \). Here, \( \mathbf{b} = \langle -1, 2, 5 \rangle \) and \( \mathbf{c} = \langle 3, 6, -1 \rangle \).

Step 3: Calculate Each Product

Compute the product of corresponding components: \(-1 \times 3 = -3\), \(2 \times 6 = 12\), and \(5 \times -1 = -5\).

Step 4: Sum the Products

Add the results of the individual products: \(-3 + 12 + (-5) = 4\). This gives us the value of the dot product \( \mathbf{b} \cdot \mathbf{c} \).

Key Concepts

Vectors

Vectors are fundamental entities in mathematics and physics that describe both a magnitude and a direction. They are usually represented in a two or three-dimensional space using coordinates. For instance, a vector can be written as \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), where \( v_1, v_2, \) and \( v_3 \) are its components along the x, y, and z axes respectively.

These components tell us how far the vector extends in each direction. Vectors are crucial in representing physical quantities such as velocity, force, and displacement. In geometry, vectors help define points, lines, and planes with ease.

Understanding vectors involves:

• Direction - indicated by the angle it makes with an axis.
• Magnitude - calculated by the length of the vector using the formula \( \sqrt{v_1^2 + v_2^2 + v_3^2} \).

By mastering vectors, one can solve many real-world problems and perform calculations in physics, engineering, and computer graphics.

Vector Components

Vector components are the projections of a vector along the axes of the coordinate system. For the vector \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), \( v_1 \), \( v_2 \), and \( v_3 \) represent the vector’s magnitude in the x, y, and z directions respectively.

These components are crucial for calculating vector operations such as addition, subtraction, and the dot product. By analyzing each component individually, complex vector calculations become simpler and more manageable.

When vectors are decomposed into components, it allows us to:

• Understand how a vector influences different dimensions.
• Easily add or subtract vectors by handling each component separately.
• Calculate vector projections and transformations within a given coordinate system.

In practical applications, breaking vectors into components helps engineers and physicists to analyze forces and movements in mechanics efficiently.

Scalar Product

The scalar product, also known as the dot product, is a key operation that combines two vectors into a single scalar quantity. This operation provides insight into how much one vector 'goes in the same direction' as another.

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

This result is a scalar (a single number), not another vector. It's computed by multiplying corresponding components of the vectors and summing those products.

The dot product has some important properties:

• It is commutative, meaning \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \).
• It is zero if the vectors are perpendicular, indicating no directional overlap.
• It scales with the magnitude of the vectors and their similarity in direction.

Understanding the dot product is essential for tackling problems related to angles between vectors, projections, and work done by a force in physics.