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Chapter 6: Problem 2

Find the magnitude of the vector (-5,-2) .

Short Answer

Expert verified
The magnitude of the vector (-5,-2) is approximately \(\sqrt{29}\) or 5.39 (rounded to two decimal places).

Step by step solution

01

Identify the components of the vector

The given vector is (-5,-2). It means the x-component is -5 and the y-component is -2.
02

Square the components

Square the x and y components of the vector: \((-5)^2 = 25\) \((-2)^2 = 4\)
03

Add the squared components

Add the squared x and y components together: \(25 + 4 = 29\)
04

Find the square root of the sum

To find the magnitude, we need to take the square root of the sum of the squared components: \(\sqrt{29} \) The magnitude of the vector (-5,-2) is approximately \(\sqrt{29}\) or 5.39 (rounded to two decimal places).

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Key Concepts

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

Vectors
Vectors are mathematical objects that have both magnitude and direction. You can think of a vector as an arrow pointing in a certain direction with a length.
They are often used in physics and engineering to represent quantities like velocity and force.

Key features of vectors include:
  • **Magnitude**: This is the length of the vector and tells you how strong or intense the vector is.
  • **Direction**: This specifies where the vector is pointing.
A vector is usually represented in component form, like \((-5, -2)\), where \(-5\) is the x-component and \(-2\) is the y-component. This tells us that the vector moves left 5 units and down 2 units in a 2D coordinate system.
Pythagorean Theorem
The Pythagorean Theorem helps us calculate the magnitude (length) of a vector by using its components. It states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

For a vector with components \(a\) and \(b\), the theorem is applied as follows:
  • First square the x-component \((-5)^2\) which results in 25.
  • Square the y-component \((-2)^2\) to get 4.
  • Add these squares: \(25 + 4 = 29\).
  • Find the square root of 29 to get \(\sqrt{29}\).
This calculation gives us the magnitude of the vector, approximately 5.39 when rounded, showing how the Pythagorean Theorem provides a simple and effective way to find vector lengths.
Component Form of Vectors
Vectors in component form help us easily perform calculations and understand directions. Each vector can be written as \((x, y)\), which represents movement along the x-axis and y-axis.

Benefits of using component form include:
  • **Clarity**: It's easy to see how far a vector moves horizontally and vertically.
  • **Mathematical Operations**: Adding, subtracting, or scaling vectors becomes straightforward.
  • **Magnitude Calculation**: With the components, you can quickly determine the vector's magnitude using the Pythagorean Theorem.
In our example, the vector \((-5, -2)\) shows a movement of 5 units left and 2 units down from the origin. Understanding component form allows us to easily model and manipulate these kinds of movements in mathematics and real-world scenarios.

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Most popular questions from this chapter

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (6+2 i)-(9-7 i) $$

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (4+2 i)+(3+8 i) $$

Show that \(\overline{z^{n}}=(\bar{z})^{n}\) for every complex number \(z\) and every positive integer \(n\).

Show that if \(z\) is a complex number, then the imaginary part of \(z\) is in the interval \([-|z|,|z|]\).

Using coordinates, show that if \(s\) and \(t\) are scalars and \(\mathbf{u}\) is a vector, then $$ (s+t) \mathbf{u}=s \mathbf{u}+t \mathbf{u} $$.
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