Problem 5 - 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. \(\mathbf{a} \cdot(4 \mathbf{b})\)

Step-by-Step Solution

Answer

\( \mathbf{a} \cdot (4\mathbf{b}) = -69 \)

Step 1: Understand the Problem

We need to find the dot product of vector \( \mathbf{a} = \langle 2, -3, 4 \rangle \) with the vector \( 4\mathbf{b} \). First, we will calculate \( 4\mathbf{b} \).

Step 2: Compute Scalar Multiplication

Multiply each component of \( \mathbf{b} = \langle -1, 2, 5 \rangle \) by 4 to obtain \( 4\mathbf{b} = 4 \times \langle -1, 2, 5 \rangle = \langle -4, 8, 20 \rangle \).

Step 3: Set Up the Dot Product

The dot product \( \mathbf{a} \cdot (4\mathbf{b}) \) is calculated using \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \(4\mathbf{b} = \langle -4, 8, 20 \rangle \).

Key Concepts

Scalar Multiplication

Scalar multiplication is an operation that involves multiplying a vector by a scalar (a single real number). This type of multiplication affects each component of the vector uniformly by the scalar value. In the context of our problem, we need to perform scalar multiplication on the vector \( \mathbf{b} = \langle -1, 2, 5 \rangle \). The scalar we are using is 4.

• Multiply each component of vector \( \mathbf{b} \) by 4.
• The calculation is as follows: \( 4 \times \langle -1, 2, 5 \rangle = \langle 4 \times -1, 4 \times 2, 4 \times 5 \rangle \).
• This results in the new vector \( \langle -4, 8, 20 \rangle \).

This is how scalar multiplication can adjust the magnitude of a vector without affecting its direction.

Dot Product

The dot product, also known as the scalar product, of two vectors results in a single scalar quantity. It provides insights into the angle between vectors and their relative orientation. In our exercise, we are asked to calculate the dot product of \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \( 4\mathbf{b} = \langle -4, 8, 20 \rangle \).

• To find the dot product, multiply corresponding components of the two vectors together.
• Then sum these products: \( 2 \times -4 + (-3) \times 8 + 4 \times 20 \).
• Specifically, this calculates to \( -8 - 24 + 80 \).
• The final answer is \( 48 \).

The dot product gives a sense of how much one vector extends in the direction of another. A positive dot product means they generally point in the same direction, while a negative result suggests they point in opposite directions.

Vectors

Vectors are essential tools in mathematics, representing quantities characterized by both magnitude and direction. They are often symbolized in component form, such as \( \mathbf{v} = \langle x, y, z \rangle \), where each component describes the vector's projection along a different axis.
In our problem:

• Vector \( \mathbf{a} = \langle 2, -3, 4 \rangle \) signifies movement in three-dimensional space.
• Vector \( \mathbf{b} = \langle -1, 2, 5 \rangle \) is another vector also in three-dimensional space.
• The components tell us how much the vector extends in each dimension.

Vectors are foundational in physics and engineering for representing forces and velocities, among other things. Understanding vector operations like scalar multiplication and dot products help manipulate and combine these quantities effectively, enabling various applications across scientific domains.