Problem 43 - Advanced Engineering Mathematics

Question

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\mathbf{b}+2(\mathbf{a}-3 \mathbf{c})\)

Step-by-Step Solution

Answer

\( \langle -11, -41, -49 \rangle \)

Step 1: Calculate the Scalar Multiplication

First, we need to compute the scalar multiplication of \( 3\mathbf{c} \). Given \( \mathbf{c} = \langle 2, 6, 9 \rangle \), multiplying by 3 gives us:\[3\mathbf{c} = 3 \times \langle 2, 6, 9 \rangle = \langle 6, 18, 27 \rangle\]

Step 2: Subtract Vectors

Now, calculate \( \mathbf{a} - 3\mathbf{c} \) using the result from Step 1. Given \( \mathbf{a} = \langle 1, -3, 2 \rangle \), perform the subtraction:\[\mathbf{a} - 3\mathbf{c} = \langle 1, -3, 2 \rangle - \langle 6, 18, 27 \rangle = \langle 1-6, -3-18, 2-27 \rangle = \langle -5, -21, -25 \rangle\]

Step 3: Multiply Difference by Scalar

Next, calculate \( 2(\mathbf{a} - 3\mathbf{c}) \). Use the result from Step 2:\[2(\mathbf{a} - 3\mathbf{c}) = 2 \times \langle -5, -21, -25 \rangle = \langle -10, -42, -50 \rangle\]

Step 4: Add Vectors

Now we add \( \mathbf{b} \) to the result from Step 3. Given \( \mathbf{b} = \langle -1, 1, 1 \rangle \), perform the addition:\[\mathbf{b} + 2(\mathbf{a} - 3\mathbf{c}) = \langle -1, 1, 1 \rangle + \langle -10, -42, -50 \rangle = \langle -1-10, 1-42, 1-50 \rangle = \langle -11, -41, -49 \rangle\]

Step 5: Final Result

The vector resulting from \( \mathbf{b}+2(\mathbf{a}-3 \mathbf{c}) \) is \( \langle -11, -41, -49 \rangle \).

Key Concepts

Scalar Multiplication

Scalar multiplication in vector algebra is a fundamental concept. It involves multiplying each component of a vector by a scalar (a constant number). For example, if you have a vector \( \mathbf{c} = \langle 2, 6, 9 \rangle \) and you want to multiply it by \( 3 \), you perform the multiplication on each element of the vector:

• Multiply the first component: \( 3 \times 2 = 6 \)
• Multiply the second component: \( 3 \times 6 = 18 \)
• Multiply the third component: \( 3 \times 9 = 27 \)

This results in a new vector \( 3\mathbf{c} = \langle 6, 18, 27 \rangle \). Scalar multiplication changes the magnitude of the vector, but it does not alter its direction unless the scalar is negative, in which case the vector reverses direction.

Vector Addition

Vector addition is another key aspect of vector algebra. It involves combining two or more vectors to form a resultant vector. This is done by adding the corresponding components of the vectors.
For instance, if you have the vectors \( \mathbf{b} = \langle -1, 1, 1 \rangle \) and \( 2(\mathbf{a} - 3\mathbf{c}) = \langle -10, -42, -50 \rangle \), the addition \( \mathbf{b} + 2(\mathbf{a} - 3\mathbf{c}) \) proceeds as follows:

• Add the first components: \( -1 + (-10) = -11 \)
• Add the second components: \( 1 + (-42) = -41 \)
• Add the third components: \( 1 + (-50) = -49 \)

The result is the vector \( \langle -11, -41, -49 \rangle \). This operation can be visualized as laying the vectors tail-to-head in a coordinate system, producing a path or resultant vector.

Vector Subtraction

Vector subtraction is akin to adding a negative vector and is performed by subtracting the components of one vector from the components of another. Subtraction involves the reverse operation of addition, meaning you subtract corresponding elements.
Let's consider the example with vectors \( \mathbf{a} = \langle 1, -3, 2 \rangle \) and \( 3\mathbf{c} = \langle 6, 18, 27 \rangle \). The subtraction \( \mathbf{a} - 3\mathbf{c} \) follows these steps:

• Subtract the first components: \( 1 - 6 = -5 \)
• Subtract the second components: \( -3 - 18 = -21 \)
• Subtract the third components: \( 2 - 27 = -25 \)

This results in the vector \( \langle -5, -21, -25 \rangle \). Unlike scalar multiplication and addition, subtraction is vector-specific and involves a change in direction and length of the resulting vector.