Problem 21 - Calculus for Biology and Medicine

Question

Use the dot product to compute the length of \([1,2,3,4]\) '.

Step-by-Step Solution

Answer

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

Step 1: Understand the Vector Length Formula

The length (or magnitude) of a vector \( \mathbf{v} = [v_1, v_2, v_3, v_4] \) is calculated using the formula: \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \). Importantly, this formula can also be derived using the dot product, where the dot product of \( \mathbf{v} \) with itself is equal to the squared length of the vector.

Step 2: Compute the Dot Product

Compute the dot product of \( \mathbf{v} \) with itself: \( \mathbf{v} \cdot \mathbf{v} = 1^2 + 2^2 + 3^2 + 4^2 \). Compute each term: \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), \( 4^2 = 16 \). Thus, \( \mathbf{v} \cdot \mathbf{v} = 1 + 4 + 9 + 16 = 30 \).

Step 3: Calculate the Length of the Vector

The length of the vector \( \mathbf{v} \) is the square root of the dot product we calculated. Therefore: \( \|\mathbf{v}\| = \sqrt{30} \).

Step 4: Simplification

\( \sqrt{30} \) is the most simplified form as \( 30 \) is not a perfect square. This is the final length of the vector.

Key Concepts

Vector Length

When we talk about the length of a vector, we're essentially referring to how long the vector is when represented in space. This concept is mathematically described as the "magnitude" of the vector. For a vector \( \mathbf{v} = [v_1, v_2, v_3, v_4] \), the vector length is determined using the formula:
\[ \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \]
This formula arises from the Pythagorean theorem extended to multiple dimensions. It gives a single positive number that represents the vector’s size, irrespective of its direction.
To compute this using the dot product, you take the dot product of the vector with itself:

• The dot product \( \mathbf{v} \cdot \mathbf{v} \) equals \( v_1^2 + v_2^2 + v_3^2 + v_4^2 \).
• Then, the vector length is the square root of this sum.

Thus, for \( [1, 2, 3, 4] \), the computation becomes \( \sqrt{1^2 + 2^2 + 3^2 + 4^2} = \sqrt{30} \), giving the vector's length.

Magnitude

Magnitude in the context of vectors is another term for a vector's length. It provides insight into the "size" or "extent" of the vector without considering its direction. Calculating the magnitude of a vector involves using the same formula as for vector length:
\[ \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \]
This value is always non-negative, as it is the result of a square root applied to a sum of squared terms, which are themselves always non-negative.
The magnitude is extremely useful because it allows us to

• Assess the vector's contribution in various applications, from physics to computer graphics.
• Compare vectors to determine which is larger or more significant in a given context.

In essence, the magnitude is all about quantifying how much of something the vector represents, solely based on its numerical values.

Vector Algebra

Vector algebra refers to the operations that can be performed on vectors, making it a fundamental part of vector mathematics and physics. Important aspects of vector algebra include dot products and the computation of vector lengths.
The dot product, an essential operation in vector algebra, involves multiplying corresponding components of two vectors and summing up these products:

• For vectors \( \mathbf{a} = [a_1, a_2, a_3, a_4] \) and \( \mathbf{b} = [b_1, b_2, b_3, b_4] \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is \( a_1b_1 + a_2b_2 + a_3b_3 + a_4b_4 \).
• It measures the cosine of the angle between the two vectors when both are normalized.

Dot products help in determining whether two vectors are orthogonal and in calculating the length of a vector as seen earlier. Through operations like these, vector algebra provides the toolkit to manipulate and understand vectors and their interactions comprehensively.