Finding The Direction Angle Of Vector V = <-5, 12>

Hey guys! Today, we're diving into a super interesting problem about vectors and their direction angles. Specifically, we’re going to figure out the approximate direction angle of the vector v = <-5, 12>. This is a classic problem in mathematics, especially in trigonometry and linear algebra, where understanding vectors and their orientations is crucial. So, let's break it down step by step and make sure we nail this concept. Before we jump into the solution, let's make sure we're all on the same page about what a direction angle actually is. In simple terms, the direction angle of a vector is the angle it makes with the positive x-axis. Think of it like this: if you were to draw the vector on a coordinate plane, the direction angle is the angle formed between the vector and the positive side of the x-axis, measured counterclockwise. This angle gives us a precise way to describe the vector's orientation in space. The direction angle is typically represented in degrees, ranging from 0° to 360°, or sometimes in radians. Knowing the direction angle is incredibly useful because it helps us understand the vector's components and how it behaves in various mathematical and physical contexts. For example, in physics, vectors are used to represent forces, velocities, and displacements. The direction angle tells us not only the magnitude of these quantities but also the direction in which they are acting. In computer graphics, vectors and their direction angles are fundamental for animations, transformations, and rendering 3D scenes. So, understanding this concept is not just about solving a math problem; it’s about grasping a fundamental tool that's used across various fields. Now that we’re clear on what a direction angle is, let’s dive into the specifics of our problem and figure out how to find the direction angle of the vector v = <-5, 12>. This will involve some basic trigonometry, so get ready to flex those math muscles!

Breaking Down the Vector v = <-5, 12>

Okay, so we have the vector v = <-5, 12>. What does this actually mean? Well, a vector in this form represents a displacement in a two-dimensional plane. The first number, -5, tells us how far to move along the x-axis, and the second number, 12, tells us how far to move along the y-axis. So, in this case, we move 5 units to the left (since it’s negative) and 12 units up. If we were to plot this on a graph, we’d start at the origin (0,0), move 5 units to the left, and then 12 units up. The vector v would then be an arrow pointing from the origin to this new point. Now, to find the direction angle, we need to figure out the angle this arrow makes with the positive x-axis. This is where trigonometry comes into play. We can think of the x and y components of the vector as forming a right-angled triangle. The x-component (-5) is the base of the triangle, the y-component (12) is the height, and the vector itself is the hypotenuse. The angle we’re interested in is the angle between the hypotenuse (the vector) and the positive x-axis. To find this angle, we can use trigonometric functions. Specifically, the tangent function is really helpful here. Remember, the tangent of an angle in a right-angled triangle is the ratio of the opposite side (the y-component) to the adjacent side (the x-component). So, in our case, tan(θ) = y/x = 12 / -5 = -2.4. This gives us the tangent of the angle, but we want the angle itself. To find the angle, we need to use the inverse tangent function, also known as arctangent or atan. So, θ = atan(-2.4). Now, here’s a little trick to keep in mind: the arctangent function will give us an angle in the range of -90° to +90°. However, our vector is in the second quadrant (since x is negative and y is positive), which means the angle we’re looking for is between 90° and 180°. So, we need to adjust the angle we get from the arctangent function to make sure it’s in the correct quadrant. This adjustment is crucial for getting the right answer. Let's calculate the arctangent of -2.4 and then see how we can adjust it to find the actual direction angle of our vector.

Calculating the Arctangent and Adjusting for the Quadrant

Alright, let's get our calculators ready! We need to find the arctangent (atan or tan⁻¹) of -2.4. When you plug that into your calculator, you should get approximately -67.38°. Now, hold on a second! Remember what we talked about earlier? The arctangent function gives us an angle between -90° and +90°, but our vector v = <-5, 12> is in the second quadrant. This means the actual direction angle should be between 90° and 180°. So, what do we do with this -67.38°? Well, we need to adjust it to fit the second quadrant. The key here is to realize that the arctangent function is giving us the angle relative to the x-axis, but in the opposite direction. Since we want the angle measured counterclockwise from the positive x-axis, we need to add 180° to our result. This is because the reference angle (the acute angle formed by the terminal side of the angle and the x-axis) is 67.38°, and we're looking for the angle that’s 180° minus this reference angle. So, our calculation looks like this: Direction Angle = 180° + (-67.38°) = 180° - 67.38° = 112.62°. Therefore, the approximate direction angle of the vector v = <-5, 12> is about 112.62°. This adjustment is super important because it ensures we're describing the direction of the vector accurately. If we had just taken the arctangent value without adjusting for the quadrant, we would have ended up with a completely different direction, which wouldn't match our vector at all. This is a common mistake that students make, so it’s crucial to always think about which quadrant your vector lies in and adjust your angle accordingly. Now that we've calculated the direction angle, let's compare it with the answer choices we have and see which one is the closest.

Comparing with Answer Choices and Wrapping Up

Okay, we've calculated the approximate direction angle of vector v = <-5, 12> to be 112.62°. Now, let's look at the answer choices provided and see which one matches up: A. 23° B. 67° C. 113° D. 157° Looking at these options, the closest one to our calculated angle of 112.62° is C. 113°. So, the approximate direction angle of vector v = <-5, 12> is 113°. Yay! We got it! This problem highlights the importance of understanding how vectors work and how to use trigonometric functions to find their direction angles. We started by defining what a direction angle is, then broke down the vector into its components, used the arctangent function to find an initial angle, and finally adjusted the angle to fit the correct quadrant. This step-by-step approach is super helpful for tackling similar problems. Remember, guys, the key to mastering these concepts is practice. Try working through other examples of finding direction angles for vectors in different quadrants. Pay close attention to the signs of the x and y components, as they will tell you which quadrant the vector is in. Also, always double-check your work and make sure your answer makes sense in the context of the problem. Understanding vectors and their direction angles is not just about acing math tests; it's a fundamental skill that’s used in many areas of science, engineering, and computer graphics. So, keep practicing, keep exploring, and you’ll become a vector pro in no time! And that's a wrap for this problem. I hope this explanation was helpful and cleared up any confusion about finding direction angles. Keep up the great work, and I’ll see you in the next math adventure!