Problem 10 - Trigonometry

Question

The magnitude of \(\mathbf{V}=\langle a, b\rangle=a \mathbf{i}+b \mathbf{j}\) is given by _____.

Step-by-Step Solution

Answer

\( |\mathbf{V}| = \sqrt{a^2 + b^2} \)

Step 1: Understand the Vector Expression

The vector \( \mathbf{V} = \langle a, b \rangle \) can also be expressed as \( a \mathbf{i} + b \mathbf{j} \), where \( a \) and \( b \) are the components of the vector along the \( x \)-axis and \( y \)-axis, respectively. Here, \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the \( x \) and \( y \) directions.

Step 2: Recall the Magnitude Formula of a Vector

The magnitude of a vector \( \mathbf{V} = \langle a, b \rangle \) is calculated by taking the square root of the sum of the squares of its components. Mathematically, it is expressed as:\[ |\mathbf{V}| = \sqrt{a^2 + b^2} \]

Step 3: Apply the Magnitude Formula

Using the magnitude formula \( |\mathbf{V}| = \sqrt{a^2 + b^2} \), we substitute the components \( a \) and \( b \) of the vector into the equation to find its magnitude.

Key Concepts

Understanding Vector Components

Vectors are vital in mathematics and physics because they have both magnitude and direction. A typical vector in a 2-dimensional space is represented as \( \mathbf{V} = \langle a, b \rangle \). This notation highlights how a vector consists of **components**, which are its projections along the coordinate axes.

• The notation \( \langle a, b \rangle \) means that the vector has a component \( a \) along the x-axis, and,
• A component \( b \) along the y-axis.

The components \( a \) and \( b \) are crucial because they determine both the direction and the magnitude of the vector.
To fully understand vectors, visualize them as arrows pointing from one position to another in space, where the components measure how far they extend in each axis. Breaking a vector into components simplifies many problems, making it possible to analyze each direction independently.

The Role of Unit Vectors

Unit vectors are like the building blocks of vectors because they provide direction but have a **magnitude** of exactly one. The two standard unit vectors in a 2-dimensional plane are \( \mathbf{i} \) and \( \mathbf{j} \), representing the unit vectors along the x-axis and y-axis, respectively.

• \( \mathbf{i} \) indicates a movement of one unit in the x-direction.
• \( \mathbf{j} \) indicates a movement of one unit in the y-direction.

When a vector is expressed as \( a \mathbf{i} + b \mathbf{j} \), it means the vector is scaled by \( a \) units in the direction of \( \mathbf{i} \) and \( b \) units in the direction of \( \mathbf{j} \).
Unit vectors thus standardize direction, allowing focus on vector length through scalar multiplication.

Calculating Vector Magnitude with the Magnitude Formula

The magnitude of a vector is an indication of how long the vector is, often akin to measuring its "length" in space. To calculate it, we utilize the **magnitude formula**. For a vector \( \mathbf{V} = \langle a, b \rangle \), the magnitude \( |\mathbf{V}| \) is determined by the formula:\[|\mathbf{V}| = \sqrt{a^2 + b^2}\]

• This formula arises from the Pythagorean Theorem, considering the vector's components \( a \) and \( b \) as legs of a right triangle.
• The magnitude is essentially the hypotenuse of this triangle.

This formula helps us see that the magnitude remains independent of the vector's angle, focusing solely on its extent or how far it stretches in space.
Understanding this calculation is essential for many applications, wherever vector quantity is relevant, such as in physics and engineering.