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Chapter 17: Problem 35

Let \(\mathbf{A}=\langle 2,4\rangle, \mathbf{B}=\langle 4,-3\rangle\) and \(\mathrm{C}=\langle-3,2\rangle\). Find \(|\mathbf{7 A}-\mathbf{B}|\)

Short Answer

Expert verified
\( |\textbf{7A} - \textbf{B}| = \sqrt{1061} \)

Step by step solution

01

- Scalar Multiplication

First, multiply vector \(\textbf{A} \) by 7. This involves multiplying each component of \(\textbf{A} \) by 7. \(\textbf{A} = \langle 2, 4 \rangle \), so \(\textbf{7A} = 7 \times \langle 2, 4 \rangle = \langle 14, 28 \rangle \).
02

- Vector Subtraction

Next, subtract vector \(\textbf{B} = \langle 4, -3 \rangle \) from the result of step 1. This involves subtracting each corresponding component: \(\textbf{7A} - \textbf{B} = \langle 14, 28 \rangle - \langle 4, -3 \rangle = \langle 14 - 4, 28 + 3 \rangle = \langle 10, 31 \rangle \).
03

- Magnitude Calculation

Finally, find the magnitude of the resulting vector. Use the formula for the magnitude of a vector: \[ |\textbf{v}| = \sqrt{x^2 + y^2} \]. Plug in the components of \(\textbf{v} = \langle 10, 31 \rangle \): \[ | \textbf{7A} - \textbf{B} | = \sqrt{10^2 + 31^2} = \sqrt{100 + 961} = \sqrt{1061} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Multiplication
Scalar multiplication is a process where you multiply a vector by a scalar (a real number). This operation changes the magnitude of the vector but not its direction. For example, multiplying vector \(\textbf{A} = \langle 2, 4 \rangle\) by 7 means multiplying both components of \(\textbf{A}\) by 7: \(\textbf{7A} = 7 \times \langle 2, 4 \rangle = \langle 14, 28 \rangle\). Here's how you do it:

\
    Multiplying the x-component: \[7 \times 2 = 14\]
    Multiplying the y-component: \[7 \times 4 = 28\]
    <\ul>So after scalar multiplication, the new vector \(\textbf{7A}\) is \(\textbf{14, 28}\).
    Understanding this concept is crucial because it allows you to scale vectors and manipulate their lengths while retaining their original direction.
Vector Subtraction
Vector subtraction involves subtracting corresponding components of two vectors from each other. Given \(\textbf{7A} = \langle 14, 28 \rangle\) and \(\textbf{B} = \langle 4, -3 \rangle\), to find \(\textbf{7A} - \textbf{B}\) you need perform the following:

\
    Subtract the x-component of \(\textbf{B}\) from the x-component of \(\textbf{7A}\): \[14 - 4 = 10\]
    Subtract the y-component of \(\textbf{B}\) from the y-component of \(\textbf{7A}\): \[28 - (-3) = 28 + 3 = 31\]
    <\ul>Therefore, \(\textbf{7A} - \textbf{B} = \langle 10, 31 \rangle\).
    Vector subtraction is essential to understand because it allows you to determine the difference between two vectors, a fundamental operation in vector calculus.
Magnitude Calculation
The magnitude of a vector is a measure of its length. It鈥檚 calculated using the Pythagorean theorem. To find the magnitude of \(\textbf{v} = \langle 10, 31 \rangle\):

\
    Square the x-component: \[10^2 = 100\]
    Square the y-component: \[31^2 = 961\]
    Sum the squares: \[100 + 961 = 1061\]
    Take the square root of the sum: \[\textbf{|v|} = \sqrt{1061}\]
    <\ul>The final magnitude is \(\textbf{|v|} = \sqrt{1061}\).
    Understanding magnitude calculation is key because it gives you the numerical length of the vector, which is indispensable in various applications of vector analysis.

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