Problem 20 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The length of the vector \([1, 1, 3]\) is \(\sqrt{11}\).

Step 1: Understanding the Problem

We need to compute the length of the vector \([1, 1, 3]\). The length of a vector, or its magnitude, can be determined using the dot product.

Step 2: Formula for the Length of a Vector

The length of a vector \(\mathbf{v} = [a, b, c]\) is given by the formula: \[ \text{length} = \sqrt{a^2 + b^2 + c^2} \] This can also be understood as the square root of the dot product of the vector with itself.

Step 3: Compute Dot Product of the Vector with Itself

To find the dot product of the vector \([1, 1, 3]\) with itself, calculate as follows: \[ [1, 1, 3] \cdot [1, 1, 3] = 1^2 + 1^2 + 3^2 \] \[ = 1 + 1 + 9 = 11 \]

Step 4: Compute the Length of the Vector

Now, we use the result from the dot product to find the length of the vector: \[ \text{length} = \sqrt{11} \]

Key Concepts

Dot Product

The dot product is a fundamental operation involving two vectors that plays a vital role in various vector applications. To understand it simply, imagine two vectors, \(\mathbf{a} = [a_1, a_2, a_3]\) and \(\mathbf{b} = [b_1, b_2, b_3]\). The dot product, denoted as \(\mathbf{a} \cdot \mathbf{b}\), is calculated as follows: \[ \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3 \]This gives a scalar value, not another vector. The dot product is particularly useful for calculating the angle between vectors, projecting one vector onto another, and in our context, determining the magnitude of a vector. It effectively measures how much one vector goes in the direction of another.

• If the dot product is zero, the vectors are perpendicular.
• If it's positive, they point in a similar direction.
• If it's negative, they point in opposite directions.

For our vector \([1,1,3]\), computing the dot product with itself provides the foundation to find its length.

Magnitude of Vector

The magnitude, or length, of a vector is a measure of how long the vector is. Think of it like measuring the length of a line from the origin to the end point of the vector in space. The formula to find the magnitude of a vector \(\mathbf{v} = [a, b, c]\) is:
\[ \| \mathbf{v} \| = \sqrt{a^2 + b^2 + c^2} \]
This formula stems from the Pythagorean theorem, and it is used here to "unwrap" the hypotenuse of a triangle formed by components \(a, b, \) and \(c\). To find the length of a vector like \([1, 1, 3]\), we first find the dot product of the vector with itself. That gives us the sum of the squares of its components. The square root of this sum then provides the magnitude. This step-by-step process not only gives the length but also emphasizes why the dot product is helpful in these calculations. For the given vector, this results in a magnitude of \(\sqrt{11}\).

Vector Operations

Vector operations include various operations such as addition, subtraction, scalar multiplication, and particularly notable ones like the dot product and cross product. These operations allow us to manipulate vectors in many ways that are essential across physics, engineering, and mathematics.

• **Addition and Subtraction:** Involve adding or subtracting corresponding components of vectors, resulting in another vector.
• **Scalar Multiplication:** Involves multiplying each component of a vector by a scalar, changing its magnitude without affecting its direction.
• **Dot Product:** Results in a scalar, typically used for finding the magnitude or angle between two vectors.
• **Cross Product:** Results in a vector that is perpendicular to the plane containing the two original vectors, used mainly in 3D vector calculations.

Each of these operations makes vectors more versatile in solving geometrical and physical problems. Specifically, calculating the dot product of a vector with itself, as shown, is an example of using vector operations to solve practical problems by finding vector lengths.