Problem 21 - Calculus for Biology and Medicine

Question

Use the dot product to compute the length of \([1,2,3,4]^{\prime}\).

Step-by-Step Solution

Answer

The length of the vector is \( \sqrt{30} \).

Step 1: Understanding the Dot Product

The dot product (also known as the scalar product) of a vector with itself gives the sum of the squares of its components. For a vector \( \mathbf{v} = [v_1, v_2, v_3, v_4] \), the dot product \( \mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2 + v_3^2 + v_4^2 \).

Step 2: Finding the Square of Each Component

For the given vector \([1, 2, 3, 4]\), we find the squares of each component:\[1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16\]

Step 3: Summing the Squares

Add up the squares calculated in the previous step:\[1 + 4 + 9 + 16 = 30\]

Step 4: Calculating the Length

The length (magnitude) of the vector is the square root of the dot product. Therefore, calculate the square root of the sum:\[\sqrt{30}\]

Step 5: Conclusion

The length of the vector \([1, 2, 3, 4]^{\prime}\) is \( \sqrt{30} \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector calculus. It provides a way to multiply two vectors, resulting in a scalar (a single number), rather than another vector. This operation is particularly useful in various mathematical, engineering, and physics applications.
To calculate the dot product of two vectors, you multiply their corresponding components and add the results. For example, the dot product of vectors \(\mathbf{a} = [a_1, a_2, a_3]\) and \(\mathbf{b} = [b_1, b_2, b_3]\) is calculated as:

• \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\)

One crucial property of the dot product is that when you take the dot product of a vector with itself (\(\mathbf{v} \cdot \mathbf{v}\)), you get the sum of the squares of its components. This sum acts as the cornerstone for finding another essential property: the length of the vector. This self-dot product is reminiscent of the Pythagorean theorem and emphasizes the geometric nature of vectors.

Vector Length

The length (or magnitude) of a vector is a measure of how far it extends from the origin in a coordinate system. Think of it as the hypotenuse in a right-angled triangle, with the other sides representing the components of the vector.
To determine the length of a vector, such as \([1, 2, 3, 4]\), you can use the result from the vector's dot product with itself. Here's how it works:

• First, compute the dot product: \(\mathbf{v} \cdot \mathbf{v} = 1^2 + 2^2 + 3^2 + 4^2 = 30\).
• Then, calculate the square root of this sum to find the length: \(\sqrt{30}\).

This approach reflects the application of the Pythagorean theorem in multiple dimensions, offering a straightforward method to assess a vector's length no matter its dimensionality.
By understanding this concept, you can appreciate the simplicity with which vector mathematics allows you to evaluate the size or "length" of extenuating directions described by vectors.

Scalar Product

The scalar product is another term for the dot product, highlighting that the outcome of this operation is always a scalar, not a vector. This distinction is important because while vectors define direction and magnitude, scalars represent only magnitude without direction.
The scalar product is calculated precisely as the dot product. It is widely applied in physics to find work done by a force, project forces along certain directions, or analyze energy in systems. In essence, it reduces a vector relationship to a single number useful for further calculations.
This operation is also essential in computer graphics and 3D modeling, where understanding the angles and orientation of objects is crucial. A scalar product of zero indicates orthogonal vectors, meaning they are perpendicular in geometric space.
Whether used in mathematics, physics, or engineering, the scalar product remains a versatile tool in translating vector information into usable numerical data. By delving deeper into scalar products, you gain insight into how vectors interrelate and the significance of their interactions in diverse scenarios.