Problem 1 - 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. \(\mathbf{a} \cdot \mathbf{b}\)

Step-by-Step Solution

Answer

The dot product \( \mathbf{a} \cdot \mathbf{b} = 12 \).

Step 1: Write Definition of Dot Product

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is calculated using the formula: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

Step 2: Identify Components of Vectors

Identify the components of \( \mathbf{a} \) and \( \mathbf{b} \): \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \( \mathbf{b} = \langle -1, 2, 5 \rangle \). Thus, \( a_1 = 2, a_2 = -3, a_3 = 4 \) and \( b_1 = -1, b_2 = 2, b_3 = 5 \).

Step 3: Apply Dot Product Formula

Substitute the components of the vectors into the dot product formula: \[ \mathbf{a} \cdot \mathbf{b} = (2)(-1) + (-3)(2) + (4)(5) \].

Step 4: Perform Multiplications

Calculate each multiplication: \( (2)(-1) = -2 \), \((-3)(2) = -6 \), and \((4)(5) = 20 \).

Step 5: Sum the Results

Add the results of the multiplications to find the dot product: \(-2 + (-6) + 20 = 12\).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors to get a scalar. This operation is especially useful in physics and engineering for finding the angle between two vectors or determining if they are perpendicular. To compute the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), you use the formula:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

For example, given vectors \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \( \mathbf{b} = \langle -1, 2, 5 \rangle \), the calculation involves:

• \( (2)(-1) = -2 \)
• \((-3)(2) = -6 \)
• \((4)(5) = 20 \)

Adding these results, \( -2 + (-6) + 20 = 12 \), gives the dot product of 12. This result is a scalar, which means it is simply a single number, not a vector.

Vectors

In mathematics and physics, vectors are essential tools. A vector is a quantity that has both magnitude and direction. It is represented as an ordered list \(\langle a_1, a_2, a_3 \rangle\), where each element is a component of the vector along a specific axis. Vectors can be visualized as arrows in space:

• The length of the arrow represents the magnitude.
• The direction of the arrow represents the direction of the vector.

Vectors are often used to model physical quantities such as velocity, force, and displacement. In our example, vectors \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \( \mathbf{b} = \langle -1, 2, 5 \rangle \) each has three components, corresponding to dimensions in 3D space. Understanding vectors helps in performing various operations like addition, subtraction, and finding dot and cross products.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation scales the vector by the given scalar. If you have a vector \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \) and a scalar \( c \), the result of the multiplication is a new vector:

• \( c \mathbf{v} = \langle cv_1, cv_2, cv_3 \rangle \)

This operation changes the magnitude of the vector but does not affect its direction unless the scalar is negative, which flips the direction. For instance, for a vector \( \mathbf{a} = \langle 2, -3, 4 \rangle \), multiplying it by a scalar 3 gives:

• \( 3 \mathbf{a} = \langle 3 \cdot 2, 3 \cdot (-3), 3 \cdot 4 \rangle = \langle 6, -9, 12 \rangle \)

This operation is fundamental in vector calculus and provides the means to stretch or shrink vectors without changing their direction.