Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the dot product to find the magnitude of \(\mathbf{u}.\) $$\mathbf{u}=20 \mathbf{i}+25 \mathbf{j}$$

Step-by-Step Solution

Answer

The magnitude of \(\mathbf{u}\) is \(\sqrt{20^2 + 25^2}\), which is equal to 32.01562118716424.

Step 1: Identify the Components of the Vector

First, identify the components of the vector. In vector \(\mathbf{u} = 20\mathbf{i} + 25\mathbf{j}\), the components are 20 (for \(\mathbf{i}\), or the x-component) and 25 (for \(\mathbf{j}\), or the y-component).

Step 2: Apply the Dot Product

Next, apply the dot product on the vector with itself. The dot product is the sum of the product of corresponding elements of two vectors. So, the dot product of \(\mathbf{u}\) with itself is \((20^2 + 25^2)\).

Step 3: Calculate the Magnitude

Finally, the magnitude of a vector is the square root of the dot product of the vector with itself. Therefore, take the square root of \((20^2 + 25^2)\) to find the magnitude of \(\mathbf{u}\).

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It plays a crucial role in a variety of applications, from physics to computer graphics. At its core, the dot product of two vectors is a scalar value, which means it is just a single number, rather than another vector.

• The dot product is calculated by multiplying the corresponding components of two vectors and summing the results.
• For example, for two vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}\), the dot product \(\mathbf{a} \cdot \mathbf{b}\) is: \(a_1b_1 + a_2b_2 + a_3b_3\).
• In the case of our vector \(\mathbf{u} = 20\mathbf{i} + 25\mathbf{j}\), the dot product with itself is \(20^2 + 25^2\).

This product represents how much one vector goes in the same direction as another, which is why it's extremely useful in calculating the magnitude of a vector.

Vector Components

Understanding vector components is essential, as they describe the vector in terms of its directions and contributions along different axes. Each vector can be broken down into components along the coordinate axes, typically labeled as \(\mathbf{i}, \mathbf{j}, \) and \(\mathbf{k}\) for the x, y, and z axes respectively.

• The x-component of a vector is the part that lies along the x-axis, the y-component along the y-axis, and so on.
• For vector \(\mathbf{u} = 20\mathbf{i} + 25\mathbf{j}\), 20 is the component along the x-axis, and 25 is the component along the y-axis.

These components allow us to perform various operations like addition, subtraction, and importantly, the dot product, which is vital for calculating vector magnitudes.

Magnitude Calculation

Finding the magnitude of a vector involves determining its length, which in essence, translates to calculating how far it extends in space. This can be crucial for understanding forces, velocities, and distances in physics and engineering.

• One straightforward method to determine the magnitude is by using the dot product of the vector with itself.
• First, calculate the dot product of the vector with itself, for \(\mathbf{u}\), it is \(20^2 + 25^2\).
• Then, take the square root of this result to get the magnitude.

In the case of vector \(\mathbf{u}\), the dot product with itself results in \(20^2 + 25^2 = 400 + 625 = 1025\). Thus, the magnitude is \(\sqrt{1025}\). This step ensures you get the length of the vector, a vital piece of information for any further vector calculations.