Problem 6 - 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. \(\mathbf{b} \cdot(\mathbf{a}-\mathbf{c})\)

Step-by-Step Solution

Answer

The scalar product is 8.

Step 1: Calculate \( \mathbf{a} - \mathbf{c} \)

Subtract the vector \( \mathbf{c} = \langle 3, 6, -1 \rangle \) from \( \mathbf{a} = \langle 2, -3, 4 \rangle \) component-wise.\[\mathbf{a} - \mathbf{c} = \langle 2 - 3, -3 - 6, 4 - (-1) \rangle = \langle -1, -9, 5 \rangle\]

Step 2: Calculate \( \mathbf{b} \cdot (\mathbf{a} - \mathbf{c}) \)

Perform the dot product of vector \( \mathbf{b} = \langle -1, 2, 5 \rangle \) and the result from Step 1, \( \mathbf{a} - \mathbf{c} = \langle -1, -9, 5 \rangle \). This is calculated by multiplying corresponding components and summing up: \[\mathbf{b} \cdot (\mathbf{a} - \mathbf{c}) = (-1)(-1) + (2)(-9) + (5)(5) = 1 - 18 + 25 = 8\]

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector mathematics. It's a way to multiply two vectors resulting in a single scalar value. This operation combines the products of each component pair corresponding to the vectors and sums them up.
To perform a dot product for vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), apply the formula:

• Multiply corresponding components: \( a_1 \times b_1, a_2 \times b_2, \) and \( a_3 \times b_3 \).
• Add these products together: \( (a_1 \times b_1) + (a_2 \times b_2) + (a_3 \times b_3) \).

In the original exercise, the dot product involves vector \( \mathbf{b} = \langle -1, 2, 5 \rangle \) and the difference vector \( \mathbf{a} - \mathbf{c} = \langle -1, -9, 5 \rangle \). The calculation is performed by multiplying \(-1 \times -1\), \(2 \times -9\), and \(5 \times 5\), and then adding these results to get 8. Thus, the scalar value obtained is 8.
The dot product provides important insights, such as the angle between two vectors. If the dot product is zero, it indicates that the vectors are orthogonal.

Vector Subtraction

Vector subtraction is a straightforward component-wise operation where each corresponding component of one vector is subtracted from another. Say you have two vectors, \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{c} = \langle c_1, c_2, c_3 \rangle \). To find the difference \( \mathbf{a} - \mathbf{c} \), you would:

• Subtract each pair of corresponding components: \( a_1 - c_1, a_2 - c_2, \) and \( a_3 - c_3 \).

Applying vector subtraction allows us to determine the vector pointing from the tip of \( \mathbf{c} \) to the tip of \( \mathbf{a} \), essentially measuring the difference or distance between the two.
In the exercise, the subtraction is performed for \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and \( \mathbf{c} = \langle 3, 6, -1 \rangle \). The result of this component-wise subtraction is \( \langle 2-3, -3-6, 4-(-1) \rangle \) which simplifies to \( \langle -1, -9, 5 \rangle \).
Understanding vector subtraction is essential for operations like finding the resultant vector or when calculating vector differences in physical scenarios.

Component-wise Operations

Component-wise operations are the bread and butter of vector mathematics. These involve performing arithmetic operations on each corresponding component of two vectors, resulting in another vector. This method ensures that operations applied to vectors maintain the data structure integrity, producing meaningful results.
For instance, if you are given two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), component-wise operations include:

• Addition: \( \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \)
• Subtraction: \( \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle \)
• Multiplication (used in element-wise operations): \( \langle a_1 \times b_1, a_2 \times b_2, a_3 \times b_3 \rangle \)

In the context of the original exercise, component-wise subtraction was used to compute \( \mathbf{a} - \mathbf{c} = \langle -1, -9, 5 \rangle \). Subsequently, the dot product is another form of component-wise multiplication, although it sums the results into a scalar.
Comprehending these operations is fundamental for exploring more complex vector operations such as cross products and understanding vector fields.