Problem 29 - Trigonometry

Question

Find the magnitude of each of the following vectors. $$\langle 0,5\rangle$$

Step-by-Step Solution

Answer

The magnitude of the vector \( \langle 0,5 \rangle \) is 5.

Step 1: Identify the components of the vector

The given vector is \( \langle 0, 5 \rangle \). This vector has two components: \( 0 \) and \( 5 \).

Step 2: Use the magnitude formula for 2D vectors

The formula to find the magnitude of a 2D vector \( \langle a, b \rangle \) is \( \sqrt{a^2 + b^2} \). For our vector, this means we will calculate \( \sqrt{0^2 + 5^2} \).

Step 3: Calculate the squares of the components

Calculate \( 0^2 \) which equals \( 0 \), and \( 5^2 \) which equals \( 25 \).

Step 4: Sum the squares of the components

Add \( 0 \) and \( 25 \) to get \( 25 \).

Step 5: Take the square root of the sum

Take the square root of \( 25 \) to find the magnitude: \( \sqrt{25} = 5 \).

Key Concepts

2D Vectors

In the world of mathematics and physics, vectors are essential quantities that contain both magnitude and direction. A 2D vector, as the name suggests, resides in a two-dimensional plane, typically represented as \( \langle a, b \rangle \), where \( a \) and \( b \) are the vector's components along the horizontal and vertical axes, respectively. These components are defined in the Cartesian coordinate system. Understanding 2D vectors is crucial because they simplify the representation of complex quantities, such as force and velocity.Dealing with 2D vectors involves various operations, such as addition, subtraction, and finding their magnitude. Visualizing a 2D vector on a graph can help by showing its point of origin, direction, and end point. For example, the vector \( \langle 0, 5 \rangle \) starts from the origin (0,0) and points straight up to (0,5). This characteristic representation highlights both components' effect on the vector’s direction in a two-dimensional space.

Magnitude Formula

The magnitude of a vector is a measure of its length. It's the distance between the vector's starting point and its end point in a coordinate plane. For a 2D vector, the magnitude formula is derived from the Pythagorean theorem. Given a 2D vector \( \langle a, b \rangle \), its magnitude \( ||\mathbf{v}|| \) is calculated using the formula:\[||\mathbf{v}|| = \sqrt{a^2 + b^2}\]This formula essentially involves squaring each component of the vector \( a \) and \( b \), summing these squares, and then taking the square root of the result. It provides a precise length of the vector in its respective plane.Applying this to our example vector \( \langle 0, 5 \rangle \):

• Calculate \( 0^2 = 0 \)
• Calculate \( 5^2 = 25 \)
• Find the sum: \( 0 + 25 = 25 \)
• Compute the square root: \( \sqrt{25} = 5 \)

This shows that the magnitude of the vector is 5 units.

Vector Components

Vector components are the building blocks of a vector, indicating its projection along the axes of the coordinate system. Each vector in 2D is composed of two components. These components can be thought of as the horizontal and vertical parts of the vector. For example, in vector \( \langle 0, 5 \rangle \), the first component \( 0 \) tells us there is no movement along the horizontal \( x \)-axis, while the second component \( 5 \) indicates a movement of 5 units along the vertical \( y \)-axis.To discern the impact of each component:

• The horizontal component \( a \) represents the vector's extension left or right.
• The vertical component \( b \) represents the vector's extension up or down.

Having a clear grasp of these components helps in not just computing the vector's magnitude, but also understanding its direction, which is essential for vector addition, subtraction, and other vector operations. Thus, knowing vector components is critical in breaking down vector problems into simpler parts.