Problem 59 - 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 \(\left[0,360^{\circ}\right)\). $$\langle- 6,0\rangle$$

Step-by-Step Solution

Answer

Magnitude: 6, Direction Angle: 180°.

Step 1: Calculate the Magnitude

To find the magnitude of vector \( \langle -6, 0 \rangle \), apply the formula \( \sqrt{x^2 + y^2} \), where \( x = -6 \) and \( y = 0 \). Substitute the values: \( \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6 \). Thus, the magnitude is 6.

Step 2: Determine the Direction Angle

To find the direction angle \( \theta \) for the vector \( \langle -6, 0 \rangle \), use the formula \( \tan \theta = \frac{y}{x} \). Since \( y = 0 \) and \( x = -6 \): \( \tan \theta = \frac{0}{-6} = 0 \). This means \( \theta \) is along the x-axis. Since \( x \) is negative, the direction angle is 180°.

Key Concepts

Vector Magnitude

When dealing with vectors, one of the first concepts to grasp is vector magnitude. The magnitude of a vector represents its length or size in the coordinate plane. It's essentially a measure of how much of something a vector represents, like a distance or speed. To calculate the magnitude of a vector, represented by coordinates \(\langle x, y\rangle\), you use the formula \( \sqrt{x^2 + y^2} \). This formula comes from the Pythagorean theorem, applied in the context of vector mathematics. For example, the vector \(\langle -6, 0\rangle\) has a magnitude of \(\sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6\). It tells us that this vector, though lying entirely on the x-axis, has a length of 6 units.

Direction Angle

The direction angle of a vector is the angle it makes with the positive x-axis, which helps show where the vector is pointing in a two-dimensional space. This angle is especially crucial for understanding the vector's orientation relative to the standard coordinate axes. To find this angle \(\theta\), you use the tangent function from trigonometry: \( \tan \theta = \frac{y}{x} \). Thus, for the vector \(\langle -6, 0\rangle\), we calculate \( \tan \theta = \frac{0}{-6} = 0\). A tangent of zero indicates that the vector is aligned with the x-axis, pointing in the direction dictated by the sign of the x-component. Since \(x = -6\), which is negative, the vector points to the left, corresponding to a direction angle of 180°. This angle places the vector straight to the left of the origin, on the negative side of the x-axis.

Trigonometry

Trigonometry is integral in vector calculations, allowing you to relate angles to side lengths in right triangles. It provides essential tools for understanding and solving problems related to vectors. In our context, trigonometric functions like sine, cosine, and tangent are used to find the angle aspects of vectors. When determining the direction angle, the tangent formula \( \tan \theta = \frac{y}{x} \) translates a vector's components directly into an angle measurement. This method is simple to apply once you break down the vector into its x and y parts and understand the unit circle's role. For vectors aligned perfectly along the axes, such calculations highlight the power of trigonometry in linking algebraic vector expressions to geometric intuitions. Mastery of these fundamental concepts makes more complex vector operations and transformations easier to understand and apply.