Problem 22 - Calculus for Biology and Medicine

Question

Use the dot product to compute the length of \([1,2,3,0]\).

Step-by-Step Solution

Answer

The length of the vector is \( \sqrt{14} \).

Step 1: Understand the Length Formula

The length (or magnitude) of a vector \( \mathbf{v} = [v_1, v_2, v_3, v_4] \) is calculated using the formula \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \). To find the length of the vector \([1,2,3,0]\), we will use this formula.

Step 2: Perform the Dot Product

The dot product of a vector \( \mathbf{v} \) with itself is \( \mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2 + v_3^2 + v_4^2 \). Substituting for our vector \([1,2,3,0]\), we compute \( 1^2 + 2^2 + 3^2 + 0^2 \).

Step 3: Calculate the Squares

Compute the squares of each component: \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), and \( 0^2 = 0 \). Hence, the dot product is \( 1 + 4 + 9 + 0 = 14 \).

Step 4: Apply the Square Root

To find the length of the vector, take the square root of the dot product result: \( \sqrt{14} \).

Step 5: Conclude with the Vector Length

The length of the vector \([1,2,3,0]\) is \( \sqrt{14} \).

Key Concepts

Dot Product

The dot product is a fundamental concept in vector calculus, often used to find various properties including the magnitude of a vector. It's a way to multiply two vectors, resulting in a scalar value. Simplifying, if we have a vector \( \mathbf{v} = [v_1, v_2, v_3, v_4] \), the dot product of \( \mathbf{v} \) with itself is calculated as \( \mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2 + v_3^2 + v_4^2 \).
This particular operation is useful because it condenses a vector into a single number. The dot product is especially helpful in determining the length, or magnitude, of the vector because it helps consolidate the computation into manageable steps.

• Computational Efficiency: Breaking down vector calculations into a simple scalar.
• Application: Used broadly in physics and engineering for resolving vector quantities.

Understanding how to compute the dot product eases much of the later work in vector calculus.

Magnitude of a Vector

The magnitude of a vector, also known as its length, is a measure of how "long" the vector is. This is crucial in fields like physics, where vectors represent forces or velocities. For a vector \( \mathbf{v} = [1,2,3,0] \), the magnitude is calculated using the formula \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \).
To find the magnitude:

• Perform the Dot Product: Compute \( 1^2 + 2^2 + 3^2 + 0^2 = 14 \).
• Calculate the Square Root: \( \sqrt{14} \), which gives the length.

When we talk about distance, whether it's in pure mathematics or applied sciences, we're often referring to the magnitude of a vector. This offers a practical method to describe how much something is moving or being displaced.

• Versatile Application: Vital in physics to describe speed and direction.
• Mathematical Clarity: Provides a clear metric for comparison.

Vector Calculus

Vector calculus extends the principles of vectors to calculus, allowing for more advanced operations involving vector fields and gradients. Amidst its many applications, one of the first steps is understanding basic vector properties such as the dot product and magnitude. These form the foundational tools needed to delve into richer territories like vector differentiation and line integration.

• Fundamental Techniques: Dot product and magnitude as stepping stones.
• Advanced Operations: Gradients, divergence, and curl are built on these basics.

In many ways, vector calculus is a bridge between linear algebra and calculus, shedding light on problems that involve multi-dimensional change. By learning about the dot product and magnitude, students prepare themselves for more complex problems in engineering and physics, where vector fields often dictate outcomes.

• Real-world Application: Crucial for modern engineering methods.
• Complex Problem Solving: Offers solutions where traditional calculus falls short.