Problem 47 - Precalculus Mathematics for Calculus

Question

Find the magnitude and direction (in degrees) of the vector. $$\mathbf{v}=\langle 3,4\rangle$$

Step-by-Step Solution

Answer

Magnitude: 5; Direction: \(53.13^\circ\).

Step 1: Calculate the Magnitude

To find the magnitude of the vector \(\mathbf{v}=\langle 3,4\rangle\), we use the formula for the magnitude of a vector: \(\|\mathbf{v}\| = \sqrt{x^2 + y^2}\), where \(x\) and \(y\) are the components of the vector. Here, \(x = 3\) and \(y = 4\). Substitute these values into the formula:\[ \|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]So, the magnitude of the vector is 5.

Step 2: Calculate the Direction Angle

To find the direction of the vector in degrees, calculated from the positive x-axis, we use the tangent function: \(\tan \theta = \frac{y}{x}\), where \(\theta\) is the angle, \(x = 3\), and \(y = 4\). Substitute these values:\[ \tan \theta = \frac{4}{3} \]To find \(\theta\), take the inverse tangent (arctan or \(\tan^{-1}\)):\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]Using a calculator, \(\theta \approx 53.13^\circ\).So, the direction of the vector is approximately \(53.13^\circ\) measured from the positive x-axis.

Key Concepts

Vector Components

Every vector in the plane can be described by its components. For a vector \( \mathbf{v} = \langle x, y \rangle \), the components \( x \) and \( y \) indicate how much the vector moves in the horizontal and vertical directions, respectively. These components form the building blocks of any vector in two dimensions.
For example, in the exercise, the vector \( \mathbf{v} = \langle 3, 4 \rangle \), the component \( x = 3 \) means you move 3 units along the x-axis, while \( y = 4 \) means you move 4 units along the y-axis. This gives the vector a clear direction and a set length on the coordinate plane.
Understanding vector components is crucial as they allow you to break down more complex vector calculations into simpler parts:

• Addition: When adding vectors, simply add their respective components.
• Subtraction: Subtract the components to find the difference between vectors.
• Scaling: Multiply each component by a scalar to stretch or shrink the vector.

Vector components help visualize how vectors operate in geometry and physics, making complex ideas more approachable.

Magnitude Calculation

The magnitude of a vector describes its length. To calculate it, use the formula given by \( \|\mathbf{v}\| = \sqrt{x^2 + y^2} \), applicable to any vector with components \( x \) and \( y \).
Think of this calculation like finding the length of the hypotenuse in a right-angled triangle, where \( x \) and \( y \) are the other two sides. This comes directly from the Pythagorean Theorem. In the example vector \( \mathbf{v} = \langle 3, 4 \rangle \), the magnitude is:
\[ \| \mathbf{v} \| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
This reflects the vector's direct distance from the origin. Magnitude provides insight into the vector's strength or length, irrespective of its direction. It's essential in applications like physics, where determining the size of a force is critical.
Moreover, magnitude highlights the scale of a vector, offering a clear measure that can be easily compared against other vectors.

Direction Angle

The direction angle of a vector is an important aspect as it tells you the vector’s orientation in the plane, typically measured from the positive x-axis.
To determine this angle, use the formula \( \tan{\theta} = \frac{y}{x} \), which arises from basic trigonometry, where \( \theta \) is the angle, and \( x \) and \( y \) are the vector's components.

• Given \( \mathbf{v} = \langle 3, 4 \rangle \), substitute values: \( \tan{\theta} = \frac{4}{3} \).
• Use inverse trigonometric functions to solve for \( \theta \): \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \).

Using a calculator, you find \( \theta \approx 53.13^\circ \).
This direction angle is vital for understanding where the vector points in two-dimensional space. It helps to visualize or map out forces and velocities, especially when combining multiple vectors. With vectors, the direction ensures we aren't just talking about how much of something there is but where it’s going as well.