Problem 31 - Trigonometry

Question

Find the magnitude of each of the following vectors. $$\mathbf{U}=5 \mathbf{i}+12 \mathbf{j}$$

Step-by-Step Solution

Answer

The magnitude of vector \( \mathbf{U} \) is 13.

Step 1: Understand the Vector Components

The vector \( \mathbf{U} = 5 \mathbf{i} + 12 \mathbf{j} \) is expressed in terms of its components along the i and j unit vectors, where 5 is the component along the x-axis, and 12 is the component along the y-axis.

Step 2: Use the Magnitude Formula for Vectors

The magnitude of a vector \( \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} \) can be found using the formula: \( ||\mathbf{a}|| = \sqrt{a_x^2 + a_y^2} \).

Step 3: Plug in the Values into the Formula

Substitute \( a_x = 5 \) and \( a_y = 12 \) into the magnitude formula: \[ ||\mathbf{U}|| = \sqrt{5^2 + 12^2} \].

Step 4: Calculate the Squares of the Components

Calculate \( 5^2 = 25 \) and \( 12^2 = 144 \).

Step 5: Sum the Squares

Add the two squares together: \( 25 + 144 = 169 \).

Step 6: Compute the Square Root

Find the square root of 169 to get the magnitude: \( \sqrt{169} = 13 \).

Key Concepts

Vector Components

Vectors are essential in physics and engineering, and understanding their components is crucial. A vector is a quantity that has both magnitude (or length) and direction. Typically, vectors in a two-dimensional plane are expressed using two perpendicular components. These components are usually labeled with unit vectors, like \( \mathbf{i} \) and \( \mathbf{j} \), which point in the direction of the x-axis and y-axis, respectively.

Let's take an example vector \( \mathbf{U} = 5 \mathbf{i} + 12 \mathbf{j} \). Here, 5 is the x-component of the vector, and it tells us how far the vector stretches along the x-axis. Similarly, 12 is the y-component, showing the extent along the y-axis. Knowing these components allows us to visualize the direction and span of the vector on a coordinate plane.

You can think of these components as the legs of a right triangle, where the vector \( \mathbf{U} \) acts like the hypotenuse. To sum up, the components provide a straightforward way to break down and analyze a vector's influence in multi-dimensional space.

Magnitude Formula

Every vector has a magnitude, representing its length, irrespective of direction. To find the magnitude, you can use the Pythagorean theorem, similar to how you determine the length of the hypotenuse in a right triangle. The formula for a vector \( \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} \) is given as:

\[||\mathbf{a}|| = \sqrt{a_x^2 + a_y^2}\]

This formula helps calculate the vector's magnitude by using its components. For instance, if you have a vector \( \mathbf{U} = 5 \mathbf{i} + 12 \mathbf{j} \), you can determine the magnitude by substituting the components into the formula:

\[||\mathbf{U}|| = \sqrt{5^2 + 12^2}\]

This step involves making the vector's components part of a Pythagorean triplet. Squaring and summing these components gives 25 and 144, resulting in a total of 169. Finally, taking the square root of 169 provides the magnitude, which in this example is 13. Breaking down a vector to its components and applying this formula makes it easy to gauge its size.

Unit Vectors

In physics and vector mathematics, unit vectors play a pivotal role. They are vectors with a magnitude of one and serve as directional guides in space. The unit vectors in two dimensions are \( \mathbf{i} \) and \( \mathbf{j} \), directed along the x-axis and y-axis, respectively.

Unit vectors don't have any width or length beyond 1 because of their defining property. Instead, they help in expressing other vectors in terms of direction. For instance, when we have \( \mathbf{U} = 5 \mathbf{i} + 12 \mathbf{j} \), the coefficients 5 and 12 indicate how far the vector stretches in each direction, using unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) as references.

Unit vectors enable easy vector manipulation and simplification in mathematical equations. They help in representing any vector concisely by breaking it down into components. Therefore, understanding unit vectors is essential for anyone dealing with vectors, as they provide a clean and efficient way to handle multi-dimensional directions.