Problem 44 - 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. \(4(\mathbf{a}+2 \mathbf{c})-6 \mathbf{b}\)

Step-by-Step Solution

Answer

The resulting vector is \( \langle 26, 30, 74 \rangle \).

Step 1: Multiply Vector c by 2

First, multiply each component of the vector \( \mathbf{c} \) by 2. If \( \mathbf{c} = \langle 2, 6, 9 \rangle \), then \[ 2 \mathbf{c} = \langle 2\cdot2, 6\cdot2, 9\cdot2 \rangle = \langle 4, 12, 18 \rangle. \]

Step 2: Add Vectors a and 2c

Next, add vectors \( \mathbf{a} \) and \( 2 \mathbf{c} \). If \( \mathbf{a} = \langle 1, -3, 2 \rangle \) and \( 2 \mathbf{c} = \langle 4, 12, 18 \rangle \), then \[ \mathbf{a} + 2 \mathbf{c} = \langle 1+4, -3+12, 2+18 \rangle = \langle 5, 9, 20 \rangle. \]

Step 3: Multiply the Resulting Vector by 4

Multiply the resulting vector from Step 2, \( \langle 5, 9, 20 \rangle \), by 4. \[ 4(\mathbf{a} + 2 \mathbf{c}) = \langle 4 \cdot 5, 4 \cdot 9, 4 \cdot 20 \rangle = \langle 20, 36, 80 \rangle. \]

Step 4: Multiply Vector b by 6

Multiply each component of vector \( \mathbf{b} \) by 6. If \( \mathbf{b} = \langle -1, 1, 1 \rangle \), then \[ 6 \mathbf{b} = \langle -1 \cdot 6, 1 \cdot 6, 1 \cdot 6 \rangle = \langle -6, 6, 6 \rangle. \]

Step 5: Subtract 6b from 4(a + 2c)

Subtract the vector we found in Step 4 from the vector in Step 3. \[ 4(\mathbf{a} + 2 \mathbf{c}) - 6 \mathbf{b} = \langle 20, 36, 80 \rangle - \langle -6, 6, 6 \rangle = \langle 20 + 6, 36 - 6, 80 - 6 \rangle = \langle 26, 30, 74 \rangle. \]

Key Concepts

Vector Addition

Vector addition is a fundamental operation in vector mathematics that involves combining two or more vectors to form a new vector. Imagine vectors as arrows pointing in a particular direction with a certain length, and vector addition as combining these arrows to find a resultant direction and length. To add vectors, you simply add their corresponding components.
For instance, if you have vectors \( \mathbf{a} = \langle 1, -3, 2 \rangle \) and \( 2 \mathbf{c} = \langle 4, 12, 18 \rangle \), you add each component from the two vectors together to form a new vector:

• Add the first components: \( 1 + 4 = 5 \).
• Add the second components: \( -3 + 12 = 9 \).
• Add the third components: \( 2 + 18 = 20 \).

This results in the new vector \( \langle 5, 9, 20 \rangle \). Vector addition is essential when working with physical quantities like force or velocity, where multiple influences need to be combined into a single effect.

Scalar Multiplication

Scalar multiplication is the process of scaling a vector by a real number, called a scalar. This operation changes the magnitude of the vector without altering its direction. Scaling a vector is like stretching or shrinking the arrow that represents it, depending on whether the scalar is greater or less than one.
For example, consider vector \( \mathbf{c} = \langle 2, 6, 9 \rangle \) from our problem. When we multiply \( \mathbf{c} \) by 2, each component of \( \mathbf{c} \) is doubled:

• First component: \( 2 \cdot 2 = 4 \)
• Second component: \( 6 \cdot 2 = 12 \)
• Third component: \( 9 \cdot 2 = 18 \)

This results in a new, larger vector \( \langle 4, 12, 18 \rangle \). Scalar multiplication is particularly useful in adjusting quantities such as velocity or acceleration, where changes in strength but not direction are needed.

Vector Subtraction

Vector subtraction is used when you need to find the difference between two vectors, essentially indicating how much one vector "falls short" of another. It is performed similarly to vector addition but involves subtracting corresponding components of the vectors.
Let's look at subtracting vector \( 6 \mathbf{b} = \langle -6, 6, 6 \rangle \) from \( 4(\mathbf{a} + 2 \mathbf{c}) = \langle 20, 36, 80 \rangle \):

• Subtract the first components: \( 20 - (-6) = 20 + 6 = 26 \)
• Subtract the second components: \( 36 - 6 = 30 \)
• Subtract the third components: \( 80 - 6 = 74 \)

The resulting vector is \( \langle 26, 30, 74 \rangle \). Vector subtraction is crucial in finding relative quantities in physics or engineering, such as displacement or difference in force, by indicating how much more or less is present compared to another vector.