Problem 19 - Calculus for Biology and Medicine

Question

Use the dot product to compute the length of \([0,-1,2]\).

Step-by-Step Solution

Answer

The length of the vector is \( \sqrt{5} \).

Step 1: Understand the Dot Product for Vector Length

To find the length of a vector \( \mathbf{v} = [x, y, z] \), we utilize the dot product \( \mathbf{v} \cdot \mathbf{v} \). The dot product for a vector with itself is equivalent to \( x^2 + y^2 + z^2 \), and the length of the vector, also known as the magnitude, is the square root of this value.

Step 2: Compute the Dot Product of the Vector with Itself

For the vector \( [0, -1, 2] \), calculate the dot product with itself: \[ [0, -1, 2] \cdot [0, -1, 2] = 0^2 + (-1)^2 + 2^2 = 0 + 1 + 4 = 5 \].

Step 3: Calculate the Length of the Vector

Take the square root of the dot product computed in Step 2 to find the length of the vector: \[ \| \mathbf{v} \| = \sqrt{5} \].

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It takes two equal-length sequences of numbers (usually vectors) and returns a single number.

• The dot product is calculated by multiplying corresponding components of two vectors and summing the results. For example, for vectors \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is \( a_1b_1 + a_2b_2 + a_3b_3 \).
• An interesting property is the dot product of a vector with itself. This simplifies to \( x^2 + y^2 + z^2 \) for a three-dimensional vector \( [x, y, z] \). When we square the length of each component and add them together, it provides a foundation for the next concept: magnitude.

Understanding the dot product is crucial when working with vectors as it offers insights into vector projections and orthogonality, which are important ideas in physics and engineering.

Magnitude of Vector

Magnitude, also known as vector length, represents how long a vector is in space. It is a concept that can be visualized as the distance from the origin to the point described by the vector in Euclidean space.

• To find the magnitude, you start with the dot product of the vector with itself, which sums the squares of its components. For a vector \( [x, y, z] \), the dot product is \( x^2 + y^2 + z^2 \).
• The magnitude of the vector is the square root of this sum, denoted as \( \| \mathbf{v} \| = \sqrt{x^2 + y^2 + z^2} \).

The magnitude gives us a scalar quantity and is always non-negative. It is a pivotal concept in physics and engineering, providing insights into physical quantities like force and velocity, which are vector quantities.

Compute Vector Length

Computing the vector length is straightforward when you know how to apply the dot product. Here’s how it works:

• First, take the dot product of the vector with itself. For example, for \([0, -1, 2]\), compute \(0^2 + (-1)^2 + 2^2 = 5\).
• Next, take the square root of the result to find the vector length. So, \( \| \mathbf{v} \| = \sqrt{5} \).

This simple method allows you to compute the distance from the origin to the point represented by the vector in a three-dimensional space efficiently. It's an essential skill, often used in applications such as computer graphics, where understanding and manipulating the size of vectors is crucial.