Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle 0,-12\rangle$$

Step-by-Step Solution

Answer

Magnitude is 12, direction angle is 270°.

Step 1: Understanding Vectors and Magnitude

A vector \( \langle a, b \rangle \) represents a direction and distance in a coordinate system. The magnitude (or length) of this vector \( \langle a, b \rangle \) is determined using the formula \( \sqrt{a^2 + b^2} \). For vector \( \langle 0, -12 \rangle \), we will substitute \( a = 0 \) and \( b = -12 \) to find the magnitude.

Step 2: Calculating the Magnitude

Substitute \( a = 0 \) and \( b = -12 \) into the magnitude formula:\[ \text{Magnitude} = \sqrt{0^2 + (-12)^2} = \sqrt{0 + 144} = \sqrt{144} = 12 \]

Step 3: Understanding Direction Angle

The direction angle of a vector \( \langle a, b \rangle \) is found using the tangent inverse function, \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). However, if \( a = 0 \), the vector points either directly up or down. For the case \( \langle 0, -12 \rangle \), the vector points straight downwards.

Step 4: Calculating the Direction Angle

Since the vector \( \langle 0, -12 \rangle \) points directly down, it is along the negative y-axis. This corresponds to a direction angle of \( 270^\circ \), because starting from the positive x-axis and moving counter-clockwise, 270 degrees brings us to the negative y-axis direction.

Key Concepts

Magnitude

In the realm of vectors, the magnitude is akin to measuring the length of an arrow. This arrow points from the starting position, often the origin in a coordinate system, to the endpoint marked by the vector itself. To find the magnitude of a vector, you use the formula:

• \[\text{Magnitude} = \sqrt{a^2 + b^2}\]

This is derived from the Pythagorean theorem. If we have a vector \( \langle a, b \rangle \), it can be thought of as a right triangle's hypotenuse, where \( a \) and \( b \) are the legs. For our specific vector \( \langle 0, -12 \rangle \), substituting directly leads us to:

• \( a = 0 \)
• \( b = -12 \)
• \( \text{Magnitude} = \sqrt{0^2 + (-12)^2} = \sqrt{144} = 12 \)

So, the magnitude of this vector is simply 12, meaning its length is 12 units when represented in a coordinate system.

Direction Angle

A vector's direction angle tells us where the arrow of the vector is pointing relative to the horizontal axis (the positive x-axis). Normally, this is calculated using the inverse tangent function:

• \[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]

However, when \( a = 0 \), we need to take a special note, because the inverse tangent would imply division by zero. In such cases:

• If \( b > 0 \), the direction is straight up at \( 90^\circ \).
• If \( b < 0 \), like in \( \langle 0, -12 \rangle \), it points directly down at \( 270^\circ \).

Directional angles always originate from the positive x-axis and measure counterclockwise, ensuring they fit within the range \([0, 360^\circ)\). Thus, for our vector, with its downward direction, the angle is \( 270^\circ \), pointing perfectly along the negative y-axis.

Coordinate System

When understanding vectors, the coordinate system is like the map on which our vector arrows are drawn. Typically, we use the Cartesian coordinate system, featuring two axes - x-axis (horizontal) and y-axis (vertical), intersecting at the origin \((0, 0)\). This system helps define where a vector's starting point is located and where it points.

• A vector \( \langle a, b \rangle \) uses the coordinates \( a \) and \( b \) to describe its direction and position.
• For \( \langle 0, -12 \rangle \), the vector starts at the origin and points exactly down the y-axis, crossing the point \((0, -12)\) on the plan.
• This layout helps in visualizing both magnitude (length) and direction (angle with respect to the x-axis).

This system enables clear representation of relationships between multiple vectors, such as when calculating resultant vectors or comparing directions and magnitudes.