Problem 14 - Precalculus

Question

Find the magnitude and direction angle of the given vector. $$\mathbf{u}=\langle 0,7\rangle$$

Step-by-Step Solution

Answer

The magnitude is 7 and the direction angle is \( 90^\circ \).

Step 1: Magnitude Formula

To find the magnitude of a vector \( \mathbf{u} = \langle a, b \rangle \), we use the formula: \[ |\mathbf{u}| = \sqrt{a^2 + b^2} \].

Step 2: Apply Magnitude Formula

Substitute \( a = 0 \) and \( b = 7 \) into the magnitude formula: \[ |\mathbf{u}| = \sqrt{0^2 + 7^2} = \sqrt{49} = 7 \].

Step 3: Direction Angle Formula

The direction angle \( \theta \) of a vector \( \mathbf{u} = \langle a, b \rangle \) is given by \( \theta = \arctan\left( \frac{b}{a} \right) \).

Step 4: Special Case Consideration

Since \( a = 0 \) and \( b \) is positive, the vector is vertical, pointing upwards. The angle \( \theta \) is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.

Key Concepts

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size. Think of it like finding the distance from the origin (0,0) to the point represented by the vector in a coordinate plane. To calculate this magnitude, we use the Pythagorean theorem. When a vector is expressed as \( \mathbf{u} = \langle a, b \rangle \), its magnitude is found using the formula:

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

In the example of the vector \( \mathbf{u} = \langle 0,7 \rangle \), the magnitude is calculated by plugging in the values for \( a \, \text{and} \, b \):

• \( |\mathbf{u}| = \sqrt{0^2 + 7^2} = \sqrt{49} = 7 \)

This tells us our vector has a magnitude of 7 units. It's a straightforward calculation indicating the overall size of the vector.

Direction Angle of a Vector

The direction angle of a vector is the angle it makes with the positive x-axis in a coordinate plane. This angle tells us in which direction the vector is pointing. If you imagine the vector as an arrow, the direction angle is essentially the angle that arrow makes as it extends from the origin. Calculating this angle can be straightforward using the tangent function.
For a vector \( \mathbf{u} = \langle a, b \rangle \), the formula to find the direction angle \( \theta \) is:

• \( \theta = \arctan\left( \frac{b}{a} \right) \)

However, certain special conditions, such as a vector lying along an axis, require special consideration. For \( \mathbf{u} = \langle 0,7 \rangle \), since \( a = 0 \), the vector is vertical and points straight up. The direction angle here is not calculated using \( \arctan \) because dividing by zero isn't defined.

• Instead, the angle is simply \( 90^\circ \) or \( \frac{\pi}{2} \text{ radians} \).

This is because the vector points directly upwards along the positive y-axis.

Vector Components

Vector components are essentially the building blocks of a vector. They are the horizontal and vertical distances from the origin to the point defined by the vector in a 2D space. Understanding vector components is like seeing a vector as a combination of two separate movements: one along the x-axis and another along the y-axis.
Mathematically, any vector \( \mathbf{u} \) in two dimensions can be described in terms of its components as \( \mathbf{u} = \langle a, b \rangle \). Here:

• \( a \) is the x-component representing the horizontal movement or displacement.
• \( b \) is the y-component representing the vertical movement or displacement.

For the vector \( \mathbf{u} = \langle 0,7 \rangle \), we see:

• The x-component is 0, indicating no movement along the x-axis.
• The y-component is 7, indicating a vertical movement upwards on the y-axis.

This breakdown into components helps us understand how the vector moves through space by separating its influence along each direction.