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Chapter 5: Problem 3

Find the length of the vector. \(\mathbf{v}=(1,2,2)\)

Short Answer

Expert verified
The length of the vector \(\mathbf{v}=(1,2,2)\) is 3.

Step by step solution

01

Identify Coordinates of the Vector

The given vector \(\mathbf{v}\) is (1,2,2). Therefore, the x-coordinate is 1, the y-coordinate is 2, and the z-coordinate is 2.
02

Apply the Formula for the Magnitude of a Vector

The formula for the magnitude (or length) of a 3D vector \(\mathbf{v}=(x,y,z)\) is \[|\mathbf{v}|=\sqrt{x^{2}+y^{2}+z^{2}}\]. Substituting the given coordinates into the formula we get: \[ |\mathbf{v}|=\sqrt{1^{2}+2^{2}+2^{2}}\]
03

Calculation

\[|\mathbf{v}|=\sqrt{1+4+4}=\sqrt{9}\]

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Key Concepts

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

3D vector
A 3D vector is a mathematical object used to define a quantity with both direction and magnitude in a three-dimensional space. Vectors are often represented using coordinates, like \(\mathbf{v}=(x, y, z)\), defining its position in space. The concept of a 3D vector is essential in fields like physics and engineering to describe things like velocities or forces.

Key components of a 3D vector include:
  • Direction: The vector points from its tail to its head in 3D space.
  • Coordinates: The vector \(\mathbf{v}=(1, 2, 2)\) has x, y, and z coordinates, where each value denotes a change along its respective axis.
  • Representation: Vectors can be graphically represented as arrows.
This makes vectors a vital tool for analyzing and visualizing complex spatial relationships.
magnitude formula
The magnitude of a vector in 3D space is a measure of its length or size. It is calculated using the magnitude formula, which is an extension of the Pythagorean theorem.

For a vector \(\mathbf{v} = (x, y, z)\), the magnitude \(|\mathbf{v}|\) is determined using the formula:
  • \[|\mathbf{v}|=\sqrt{x^{2}+y^{2}+z^{2}}\]
This formula helps to compute the straight-line distance from the origin to the point \(\mathbf{v}\) in 3D space.

In our example with \(\mathbf{v}=(1, 2, 2)\):
  • The calculation becomes \(|\mathbf{v}|=\sqrt{1^{2}+2^{2}+2^{2}}=\sqrt{9}=3\).
Understanding the magnitude allows you to quantify how much 'force' or 'movement' a vector represents.
coordinate identification
Identifying the coordinates of a vector is a fundamental step in vector analysis. It involves determining the values along each axis that define the vector's position in a given space.

For the vector \(\mathbf{v} = (1, 2, 2)\), the coordinates are:
  • x-coordinate: 1
  • y-coordinate: 2
  • z-coordinate: 2
These coordinates help to pinpoint the exact location and direction of the vector in three-dimensional space.

Coordinate identification is crucial because:
  • It forms the foundation for applying operations like dot product, cross product, and calculating length or projections.
  • It can be visualized easily in graphs or physical systems.
Mastering this skill aids in efficiently tackling more complex vector-related problems.

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Most popular questions from this chapter

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). \(f(x)=(x-\pi)^{2}, \quad\) third order

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