Calculating Dot Product Of Vectors U And V A Step By Step Guide

Hey everyone! Today, we're diving into a cool problem involving vectors and dot products. It might seem tricky at first, but trust me, we'll break it down step by step. Our main goal here is to figure out the dot product of two vectors, u and v, given some specific information about them.

The Problem at Hand

So, here's the scenario: We've got a vector u, which is defined as <1, √3>. We also know that the magnitude (or length) of vector v, denoted as |v|, is 6. And to top it off, the angle between these two vectors is 120 degrees. Our mission, should we choose to accept it (and we do!), is to calculate u · v, which represents the dot product of u and v. Sounds like a fun challenge, right?

Breaking Down the Basics of vector

Before we jump into solving the problem directly, let's quickly refresh what vectors are and what the dot product is all about. Think of a vector as an arrow in space – it has both a magnitude (length) and a direction. In our case, vector u can be visualized as an arrow starting from the origin (0,0) and pointing to the point (1, √3) on a coordinate plane. The magnitude, |v| = 6, tells us how long the arrow representing vector v is, but we don't know its exact direction yet. The angle of 120 degrees between them gives us a sense of their relative orientation. It indicates how much v is rotated relative to u. To really nail this, you gotta understand that the dot product is like a special operation that tells us how much two vectors are aligned. A larger dot product (more positive) means they're pointing in roughly the same direction, while a negative dot product means they're pointing in roughly opposite directions. A dot product of zero means they're perfectly perpendicular (at a 90-degree angle).

The Dot Product Formula

Now, let's get to the heart of the matter: the formula for calculating the dot product. There are actually two ways to calculate it, and we'll use the one that fits our given information best. The formula we're going to use is:

u · v = |u| |v| cos(θ)

Where:

• |u| is the magnitude of vector u.
• |v| is the magnitude of vector v.
• θ (theta) is the angle between the two vectors.

This formula is super handy because it directly relates the dot product to the magnitudes of the vectors and the angle between them. Let's break down each part and see how we can apply it to our problem. First up, we need to find |u|, the magnitude of vector u. Remember, u is given as <1, √3>, so we'll need to use the magnitude formula.

Calculating the Magnitude of Vector u

The magnitude of a vector is basically its length. If we have a vector in component form, like u = <a, b>, then its magnitude, |u|, is calculated using the Pythagorean theorem. Think of it as finding the hypotenuse of a right triangle where 'a' and 'b' are the lengths of the sides. The formula looks like this:

|u| = √(a² + b²)

In our case, vector u is <1, √3>, so a = 1 and b = √3. Let's plug these values into the formula:

|u| = √(1² + (√3)²) |u| = √(1 + 3) |u| = √4 |u| = 2

So, the magnitude of vector u is 2. Awesome! We've got one piece of the puzzle. Now, let's take a look at what else we know and what we still need to find out. We already know |v| is 6 (given in the problem), and we know the angle θ is 120 degrees. It seems like we have all the ingredients to calculate the dot product using our formula.

Putting It All Together: Calculating u · v

Alright, we've got all the pieces of the puzzle. We know:

• |u| = 2
• |v| = 6
• θ = 120 degrees

Now we can plug these values into the dot product formula:

u · v = |u| |v| cos(θ) u · v = (2)(6)cos(120°)

Now, we need to figure out what cos(120°) is. If you remember your unit circle or trig values, you'll know that cos(120°) = -1/2. If you're not super familiar with that, no worries! You can always use a calculator to find the cosine of 120 degrees. So, let's plug that in:

u · v = (2)(6)(-1/2) u · v = 12(-1/2) u · v = -6

And there you have it! The dot product of vectors u and v is -6. So, the correct answer is B. -6.

Wrapping Up and Key Takeaways

Okay, we did it! We successfully calculated the dot product of two vectors using the magnitude and angle formula. The key here was breaking down the problem into smaller, manageable steps. We first made sure we understood the basic concepts of vectors and the dot product. Then, we identified the formula we needed and systematically calculated each component. We found the magnitude of vector u, recalled the given magnitude of vector v, and used the angle between them to calculate the cosine. Finally, we plugged everything into the formula and arrived at our answer. This whole process highlights how important it is to have a solid understanding of the fundamental formulas and concepts. Knowing how to apply the magnitude formula and the dot product formula is crucial for solving these types of problems. And remember, practice makes perfect! The more you work with vectors and dot products, the more comfortable you'll become with them. And don't be afraid to visualize what's going on – drawing a quick sketch of the vectors can often help you understand the problem better. Understanding the relationship between the dot product, magnitudes, and the angle between vectors is super useful in many areas of math and physics. So, keep practicing, and you'll be a dot product pro in no time!

Practice Problems

To really solidify your understanding, let's look at a couple of practice problems. These will help you get more comfortable with the dot product formula and its applications.

Practice Problem 1:

Suppose vector a = <3, -4>, |b| = 5, and the angle between the vectors is 60 degrees. What is a · b?

Practice Problem 2:

Suppose vector p = <-2, 2√3>, |q| = 4, and the angle between the vectors is 150 degrees. What is p · q?

Try solving these on your own, and remember the steps we went through in the main problem. Break it down, calculate the magnitudes, find the cosine of the angle, and then plug everything into the dot product formula. You've got this!

By working through these examples and practice problems, you'll become much more confident in your ability to handle vector dot product calculations. So keep up the great work, and happy calculating!