Problem 88 - Precalculus

Question

Explain the mistake that is made. Find the direction angle of the vector \langle-2,-8\rangle Solution: \(\begin{array}{l}\text { Write the formula for the direction } \\ \text { angle of }\langle a, b\rangle .\end{array} \quad \tan \theta=\frac{b}{a}\) Let \(a=-2\) and \(b=-8\) \(\tan \theta=\frac{-8}{-2}\) Apply the inverse tangent function. \(\theta=\tan ^{-1} 4\) Evaluate with a calculator, \(\quad \theta=76^{\circ}\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was not adjusting for the vector being in the third quadrant; the corrected angle should be \( 256^{\circ} \).

Step 1: Identify the Formula for Direction Angle

To find the direction angle \( \theta \) of a vector \( \langle a, b \rangle \), we use the formula \( \tan \theta = \frac{b}{a} \). This formula relates the tangent of the angle to the ratio of the y-component \( b \) to the x-component \( a \) of the vector.

Step 2: Substitute the Vector Components

We substitute the given values \( a = -2 \) and \( b = -8 \) into the formula: \( \tan \theta = \frac{-8}{-2} \). This simplifies to \( \tan \theta = 4 \).

Step 3: Apply Inverse Tangent Function

Using the simplified ratio, we apply the inverse tangent to find \( \theta \): \( \theta = \tan^{-1}(4) \). A calculator gives \( \theta = 76^{\circ} \). Here is where the mistake happens.

Step 4: Consider the Vector's Quadrant

The vector \( \langle -2, -8 \rangle \) is in the third quadrant because both components are negative. Angles in the third quadrant are \( 180^{\circ} + \theta \). The angle should not just be \( \tan^{-1}(4) \).

Step 5: Correct the Direction Angle

Since the vector is in the third quadrant, the correct angle is given by adding \( 180^{\circ} \) to \( 76^{\circ} \), resulting in \( 256^{\circ} \).

Key Concepts

Inverse Tangent Function

The inverse tangent function, also known as arctangent, is key in finding the direction angle of vectors. By using the inverse tangent function, you reverse the process of finding a tangent from an angle, which helps in determining an angle from a tangent value. Here is how it connects to the direction angle:

• Tangent: It's the ratio of the side opposite the angle to the adjacent side in a right triangle.
• Inverse Tangent: If you have a tangent value, the inverse tangent function, denoted as \( \tan^{-1} \) or \( \arctan \), helps find the angle whose tangent is that value.

In the context of vectors, if you have a vector \( \langle a, b \rangle \), the tangent of the direction angle \( \theta \) is given by \( \frac{b}{a} \). Therefore, \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
While this calculation often shows you an angle, it's important to consider in which direction this angle is pointing on the coordinate plane.

Quadrants of Angles

Angles can be located in different quadrants of the coordinate plane, and understanding this is crucial when computing vector direction angles correctly. Here is how the quadrants work:

• Quadrant I: Both x and y components are positive. Angles are between \( 0^{\circ} \) and \( 90^{\circ} \).
• Quadrant II: X is negative, y is positive. Angles range between \( 90^{\circ} \) and \( 180^{\circ} \).
• Quadrant III: Both x and y are negative, which is crucial for our exercise. Angles span \( 180^{\circ} \) to \( 270^{\circ} \).
• Quadrant IV: X is positive, y is negative. Angles lie between \( 270^{\circ} \) and \( 360^{\circ} \).

The direction angle assumes specific adjustments based on its quadrant. For example, if the calculation gives an angle in the first quadrant, but the vector is in the third quadrant, you must add \( 180^{\circ} \) to adjust for proper location in the 360-degree system.

Vectors in Third Quadrant

Vectors pointing to the third quadrant, like \( \langle -2, -8 \rangle \), need special consideration due to their negative components. This results in a direction angle that requires adjustment.
When components are negative, as when both \( a \) and \( b \) are negative, the vector is in the third quadrant. Here’s how this affects the calculation:

• The initial calculation via inverse tangent might lead to an angle appearing to belong in either the first or second quadrant, often because of its smaller range.
• The reality is for negative x and y values, we should anticipate angles between \( 180^{\circ} \) and \( 270^{\circ} \).
• Therefore, to correctly find what the direction angle should be, it's vital to add \( 180^{\circ} \) to any angle derived from \( \tan^{-1}(\frac{-b}{a}) \) that doesn’t inherently consider quadrant placement, ensuring the angle reflects its true position.

Such adjustments ensure that when checking vector direction, we're tallying with true orientation and coordinate planes as expected by real-world applications.