Hey guys! Let's dive into the fascinating world of vectors and explore the relationship between two specific vectors: u = <-5, 1> and v = <8, 6>. We're going to figure out if these vectors point in the same direction, form an acute angle, form an obtuse angle, or point in opposite directions. So, buckle up and get ready for a vector adventure!
Understanding Vectors and Their Relationships
Before we jump into the specifics, let's refresh our understanding of vectors and how they can relate to each other. A vector, in simple terms, is a quantity that has both magnitude (length) and direction. Think of it as an arrow pointing in a certain way. Now, two vectors can have several types of relationships:
- Same Direction: Vectors pointing in the same direction are scalar multiples of each other. This means one vector can be obtained by multiplying the other vector by a positive constant.
- Opposite Direction: Vectors pointing in opposite directions are scalar multiples of each other, but this time the constant is negative.
- Acute Angle: Vectors form an acute angle if the angle between them is less than 90 degrees.
- Obtuse Angle: Vectors form an obtuse angle if the angle between them is greater than 90 degrees but less than 180 degrees.
- Orthogonal (Perpendicular): Vectors are orthogonal if the angle between them is exactly 90 degrees. This is a special case where the dot product of the vectors is zero.
To determine the relationship between vectors, we often use tools like the dot product and the angle formula. The dot product gives us information about the alignment of the vectors, while the angle formula helps us calculate the precise angle between them. These are the weapons in our arsenal as we tackle the problem at hand.
Now, let’s deep dive into how the dot product helps us understand angles. The dot product, mathematically, is a scalar value that can be calculated from two vectors. For vectors u = <u1, u2> and v = <v1, v2>, the dot product is given by: u · v = u1v1 + u2v2. But here’s where it gets interesting: the dot product is also related to the cosine of the angle (θ) between the vectors by the formula: u · v = ||u|| ||v|| cos θ, where ||u|| and ||v|| represent the magnitudes (lengths) of vectors u and v, respectively. This formula is crucial because it links the algebraic calculation of the dot product to the geometric concept of the angle between vectors. If the dot product is positive, cos θ is positive, implying the angle θ is acute (less than 90 degrees). Conversely, a negative dot product means cos θ is negative, indicating an obtuse angle (between 90 and 180 degrees). If the dot product is zero, then cos θ is zero, which means θ is 90 degrees, and the vectors are orthogonal. This connection between the dot product and the angle allows us to determine whether the angle between our vectors u and v is acute, obtuse, or right, simply by calculating the dot product and observing its sign. Isn’t that neat?
Analyzing Vectors u = <-5, 1> and v = <8, 6>
Let's apply our knowledge to the given vectors, u = <-5, 1> and v = <8, 6>. First, we'll calculate the dot product of u and v:
u · v = (-5)(8) + (1)(6) = -40 + 6 = -34
The dot product is -34, which is a negative value. Remember what we discussed earlier? A negative dot product indicates that the angle between the vectors is obtuse. So, just from this calculation, we can lean towards option C.
But to be absolutely sure, and for the sake of a thorough analysis, let's also calculate the magnitudes of u and v:
||u|| = √((-5)² + (1)²) = √(25 + 1) = √26
||v|| = √((8)² + (6)²) = √(64 + 36) = √100 = 10
Now, we can use the angle formula to find the cosine of the angle between u and v:
cos θ = (u · v) / (||u|| ||v||) = -34 / (√26 * 10) = -34 / (10√26)
Since cos θ is negative, this confirms that the angle θ is indeed obtuse. We don't even need to find the exact angle; the sign of the cosine is sufficient to tell us that the angle is obtuse. This solidifies our understanding – these vectors are definitely forming an angle greater than 90 degrees.
In addition to understanding the angle, let's also consider if the vectors point in the same or opposite directions. Vectors pointing in the same direction are scalar multiples of each other by a positive constant. For instance, if v was simply a scaled-up version of u, we could express v as ku, where k is a positive number. Similarly, vectors pointing in opposite directions are scalar multiples of each other but by a negative constant. In our case, it's clear that v is not a scalar multiple of u because there's no single number we can multiply each component of u by to get the components of v. The ratio of the x-components is -8/5, while the ratio of the y-components is 6. Since these ratios are not the same, the vectors are not scalar multiples of each other, meaning they don't point in the same or opposite directions. This is visually evident as well, as u has a negative x-component and a small positive y-component, pushing it towards the second quadrant, while v has both positive components, positioning it in the first quadrant. So, we've reinforced our understanding of the directional relationship between these vectors, making our final determination even more robust.
The Verdict
After our thorough analysis, it's clear that the correct statement is:
C. The vectors form an obtuse angle.
We've seen how the dot product, magnitudes, and the angle formula work together to reveal the relationship between vectors. It’s like being a detective, using mathematical clues to solve a vector mystery! I hope this breakdown has been helpful and insightful, guys. Keep exploring the fascinating world of vectors!
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