Finding Direction Angle Of 1/2v Where V = <-10, 15>

In the realm of vector mathematics, understanding the direction angle of a vector is crucial for various applications, ranging from physics to computer graphics. This article delves into the process of determining the direction angle of a vector, specifically focusing on the vector $\frac{1}{2}v$, where $v = \langle -10, 15 \rangle$. We will explore the underlying concepts, the steps involved in the calculation, and the significance of the result. By the end of this discussion, you will have a solid grasp of how to find the direction angle of a vector and its scalar multiples.

Core Concepts: Vectors and Direction Angles

Before we dive into the specifics, let's establish a clear understanding of the fundamental concepts. A vector is a mathematical object that has both magnitude (length) and direction. It is often represented as an ordered pair (in two dimensions) or an ordered triple (in three dimensions). For instance, the vector $v = \langle -10, 15 \rangle$ indicates a displacement of -10 units along the x-axis and 15 units along the y-axis.

The direction angle of a vector, denoted by $\theta$, is the angle formed between the vector and the positive x-axis, measured counterclockwise. This angle provides a precise way to describe the vector's orientation in the coordinate plane. Direction angles are typically expressed in degrees, ranging from 0° to 360°.

Understanding direction angles is vital because it allows us to decompose a vector into its horizontal and vertical components, which is essential for many calculations in physics and engineering. For example, when analyzing projectile motion, we need to know the initial velocity vector's direction angle to determine the range and maximum height of the projectile.

To find the direction angle $\theta$ of a vector $\langle x, y \rangle$, we often use the arctangent function ($\arctan$ or $tan^{-1}$). The tangent of the direction angle is given by the ratio of the y-component to the x-component: $tan(\theta) = \frac{y}{x}$. However, the arctangent function only returns angles in the range of -90° to 90°, so we need to consider the quadrant in which the vector lies to determine the correct direction angle.

The Significance of Quadrants

The coordinate plane is divided into four quadrants, each with its own sign conventions for x and y coordinates. This affects how we interpret the arctangent result:

• Quadrant I (x > 0, y > 0): The angle returned by the arctangent function is the correct direction angle.
• Quadrant II (x < 0, y > 0): The direction angle is 180° plus the angle returned by the arctangent function.
• Quadrant III (x < 0, y < 0): The direction angle is 180° plus the angle returned by the arctangent function.
• Quadrant IV (x > 0, y < 0): The direction angle is 360° plus the angle returned by the arctangent function (or simply subtract 360° from the angle if it is negative).

Considering the quadrant is crucial for obtaining the accurate direction angle, especially when dealing with vectors that point in directions other than the first quadrant. Failing to do so can lead to errors in calculations and misinterpretations of vector orientations.

Problem Statement: Finding the Direction Angle of $\frac{1}{2}v$

Now, let's return to the original problem. We are given the vector $v = \langle -10, 15 \rangle$ and asked to find the approximate direction angle of $\frac{1}{2}v$. This problem combines the concepts of scalar multiplication and direction angles. First, we need to find the vector $\frac{1}{2}v$. Then, we will apply the arctangent function and consider the quadrant to determine the direction angle.

Scalar multiplication is a fundamental operation in vector algebra. It involves multiplying a vector by a scalar (a real number). This operation scales the magnitude of the vector without changing its direction (if the scalar is positive) or reverses its direction (if the scalar is negative). In our case, we are multiplying the vector $v$ by the scalar $\frac{1}{2}$. This will halve the magnitude of $v$ but will not alter its direction since $\frac{1}{2}$ is a positive scalar.

The problem highlights the importance of understanding how scalar multiplication affects the direction angle of a vector. Multiplying a vector by a positive scalar does not change its direction angle, while multiplying by a negative scalar will result in a direction change of 180 degrees. This is a critical concept to keep in mind when dealing with vector operations.

Step-by-Step Solution

Let's break down the solution into manageable steps:

1. Calculate $\frac{1}{2}v$: To find $\frac{1}{2}v$, we multiply each component of $v$ by $\frac{1}{2}$:

   $$\frac{1}{2}v = \frac{1}{2} \langle -10, 15 \rangle = \langle \frac{1}{2}(-10), \frac{1}{2}(15) \rangle = \langle -5, 7.5 \rangle$$

   Thus, the vector $\frac{1}{2}v$ is $\langle -5, 7.5 \rangle$.
2. Find the Tangent of the Direction Angle: The tangent of the direction angle $\theta$ is given by the ratio of the y-component to the x-component:

   $$tan(\theta) = \frac{y}{x} = \frac{7.5}{-5} = -1.5$$

3. Use the Arctangent Function: We use the arctangent function to find the reference angle:

   $$\theta_{ref} = arctan(-1.5) \approx -56.31^{\circ}$$

   This angle is negative, indicating it is measured clockwise from the positive x-axis.
4. Determine the Quadrant: The vector $\langle -5, 7.5 \rangle$ has a negative x-component and a positive y-component, which means it lies in Quadrant II.
5. Calculate the Direction Angle: In Quadrant II, the direction angle is given by:

   $$\theta = 180^{\circ} + \theta_{ref} = 180^{\circ} + (-56.31^{\circ}) = 180^{\circ} - 56.31^{\circ} \approx 123.69^{\circ}$$

   Rounding to the nearest degree, the direction angle is approximately 124°.

Analyzing the Result

The calculated direction angle of approximately 124° aligns with our understanding of vectors and their orientations. Since the vector $\frac{1}{2}v$ lies in the second quadrant, we expect the direction angle to be between 90° and 180°. The value of 124° falls within this range, which validates our calculation.

The fact that $\frac{1}{2}v$ has the same direction angle as $v$ (before rounding, of course!) illustrates an important property of scalar multiplication. Multiplying a vector by a positive scalar changes its magnitude but not its direction. This means that the vector $\frac{1}{2}v$ points in the same direction as $v$, just with half the length. This understanding is crucial in various applications, such as physics, where forces and velocities are represented as vectors.

Conclusion

In conclusion, the approximate direction angle of $\frac{1}{2}v$, where $v = \langle -10, 15 \rangle$, is 124°. This result was obtained by first calculating the vector $\frac{1}{2}v$, then finding the arctangent of the ratio of its components, and finally adjusting the angle based on the quadrant in which the vector lies. This step-by-step process highlights the importance of understanding core concepts such as vectors, direction angles, scalar multiplication, and the significance of quadrants in vector analysis.

By mastering these concepts, you can confidently solve a wide range of problems involving vectors and their directions, which are essential in various fields, including physics, engineering, computer graphics, and more. This exploration has not only provided a solution to the specific problem but also reinforced the underlying principles of vector mathematics, enabling a deeper understanding and appreciation of the subject.