Problem 19
Question
Use the dot product to compute the length of \([0,-1,2]^{\prime}\).
Step-by-Step Solution
Verified Answer
The length of the vector \([0,-1,2]'\) is \(\sqrt{5}\).
1Step 1: Understand the Dot Product
The dot product of a vector with itself is equal to the square of its length. If we have a vector \( \mathbf{v} = [x, y, z]' \), then the dot product \( \mathbf{v} \cdot \mathbf{v} = x^2 + y^2 + z^2 \).
2Step 2: Calculate the Dot Product
Apply the formula for the dot product to the given vector \([0,-1,2]'\):\[ [0,-1,2] \cdot [0,-1,2] = 0^2 + (-1)^2 + 2^2. \] Compute each term: - \(0^2 = 0\), - \((-1)^2 = 1\), - \(2^2 = 4\). Adding these gives: \(0 + 1 + 4 = 5\).
3Step 3: Calculate the Length of the Vector
The length of the vector is the square root of the dot product. Therefore, the length is:\[ \sqrt{5} \].
Key Concepts
Dot ProductVector LengthMathematical Computation
Dot Product
The dot product is a fundamental operation in vector calculus. It occurs when you multiply two vectors, yielding a scalar result. For a vector \( \mathbf{v} = [x, y, z]' \), the dot product \( \mathbf{v} \cdot \mathbf{v} = x^2 + y^2 + z^2 \).
The dot product reveals how much of one vector points in the same direction as another. It's particularly useful to find the angle between two vectors or check perpendicularity. For instance, the dot product of a vector with itself, as in our exercise, computes a result directly related to the vector's length.
In practical terms, the dot product helps in varying fields like physics, engineering, and computer graphics. Here, we leverage it to simplify calculating vector length by converting vector components into a single, easy-to-compute number.
The dot product reveals how much of one vector points in the same direction as another. It's particularly useful to find the angle between two vectors or check perpendicularity. For instance, the dot product of a vector with itself, as in our exercise, computes a result directly related to the vector's length.
In practical terms, the dot product helps in varying fields like physics, engineering, and computer graphics. Here, we leverage it to simplify calculating vector length by converting vector components into a single, easy-to-compute number.
Vector Length
Vector length, often called the magnitude of a vector, tells you how long the vector is. Think of it as the distance from the vector's starting point to its end point in space. In a Cartesian coordinate system, for a vector \( [x, y, z]' \), the length is determined by the formula \( \sqrt{x^2 + y^2 + z^2} \).
This is why the dot product plays a pivotal role: it simplifies obtaining the squared components into one sum. Then, taking the square root of this sum, you get the length of the vector. Using our exercise with vector \( [0, -1, 2]' \), we first compute the dot product, \( 5 \), then take the square root to find the length as \( \sqrt{5} \).
Understanding vector length is key for applications such as normalization, where you scale a vector down to unit length, or verifying that two vectors are equal in length when comparing geometric shapes and angles.
This is why the dot product plays a pivotal role: it simplifies obtaining the squared components into one sum. Then, taking the square root of this sum, you get the length of the vector. Using our exercise with vector \( [0, -1, 2]' \), we first compute the dot product, \( 5 \), then take the square root to find the length as \( \sqrt{5} \).
Understanding vector length is key for applications such as normalization, where you scale a vector down to unit length, or verifying that two vectors are equal in length when comparing geometric shapes and angles.
Mathematical Computation
Mathematical computation involves using formulas and operations to solve problems, like calculating the vector length using the dot product. Here, computation requires breaking down vector components \([0, -1, 2]'\):
This process emphasizes the power of mathematical computation, streamlining complex tasks into manageable steps. It underlies many technological advancements, helping solve real-world problems efficiently. From robots calculating paths to engineers designing structures, these principles make it all possible with precision and ease. By mastering these steps, you enhance your mathematical toolkit, ready for advanced applications and problem-solving.
- The square of each component: \(0^2 = 0\), \((-1)^2 = 1\), \(2^2 = 4\).
- Sum these squared values: \(0 + 1 + 4 = 5\).
- Result gives the dot product, \(5\), aiding the calculation of vector length.
This process emphasizes the power of mathematical computation, streamlining complex tasks into manageable steps. It underlies many technological advancements, helping solve real-world problems efficiently. From robots calculating paths to engineers designing structures, these principles make it all possible with precision and ease. By mastering these steps, you enhance your mathematical toolkit, ready for advanced applications and problem-solving.
Other exercises in this chapter
Problem 19
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