Unraveling the Cross Product and Magnitude of Vectors

Unraveling the Mystery of Vectors: Cross Products Made Easy

Vectors are a fundamental part of mathematics and physics, acting as building blocks for everything from engineering to computer graphics. You may have stumbled upon questions involving vectors in various formats, but let’s take a deeper look, especially at cross products. Today, we’re going to tackle a specific example that illustrates how to find the magnitude of a resultant vector formed by the cross-product of two given vectors. Sounds daunting? Don’t worry; I assure you, it’s simpler than it seems!

The Question at Hand

Let’s set the stage with our two vectors:

• Vector a = (2, -4, 4)

• Vector b = (4, 0, 3)

The question essentially asks: What’s the magnitude of the resultant vector c formed by the cross product of vectors a and b? Your answer options are:

• A. 5√2

• B. 10√5

• C. 12

• D. 15

Now, before we jump to the conclusion, let’s explore how we arrive at the answer step-by-step.

A Quick Brush-Up on Cross Products

If you’re wondering why we use cross products in the first place, here’s a quick insight. The cross product gives us a new vector that is orthogonal to the plane formed by the two original vectors. Think of it as creating a vector that lives “above” or “below” the surface made by your chosen vectors. This is especially useful in three-dimensional space where directions matter a lot!

The formula to find the magnitude of the cross product is

[

|c| = |a \times b| = |a| \cdot |b| \cdot \sin(\theta)

]

However, in many scenarios, it’s easier to compute the determinant formed by the components of the vectors, which we’re going to do next.

Computing the Cross Product

To find the cross product, we set up a determinant using a 3x3 matrix. It looks like this:

[

c = \begin{vmatrix}

\hat{i} & \hat{j} & \hat{k} \

2 & -4 & 4 \

4 & 0 & 3

\end{vmatrix}

]

What we’re doing here is applying the method of determinants. Each of those components—𝑖, 𝑗, and 𝑘—represents the axes in a three-dimensional space that creates our new vector c. When we calculate this determinant, we get:

[

c = \hat{i}((-4)(3) - (4)(0)) - \hat{j}((2)(3) - (4)(4)) + \hat{k}((2)(0) - (-4)(4))

]

Let me break that down for you:

• For the 𝑖 component: ((-12 - 0) = -12)

• For the 𝑗 component: ((6 - 16) = -10) (notice the negative sign!)

• For the 𝑘 component: ((0 + 16) = 16)

Putting it all together, we find the components of vector c:

[

c = (-12\hat{i} + 10\hat{j} + 16\hat{k})

]

Finding the Magnitude

Now that we have our cross product, calculating the magnitude involves applying the Pythagorean theorem in three dimensions:

[

|c| = \sqrt{(-12)^2 + (10)^2 + (16)^2}

]

Calculating each term:

• ((-12)^2 = 144)

• ((10)^2 = 100)

• ((16)^2 = 256)

Adding these up gives us:

[

144 + 100 + 256 = 500

]

Finally, taking the square root of 500 yields:

[

|c| = \sqrt{500} = 10\sqrt{5}

]

Drumroll, Please!

So, what’s the magnitude of our new vector c after performing the cross product? If you guessed option B, you’re spot on—10√5 is the answer!

Why Understanding Cross Products Matters

Navigating through the world of vectors and understanding their properties like cross products is fundamental to many fields. Whether it’s physics, engineering, or computer science, these skills often come into play in real-life applications. Picture this: if you’ve ever played a video game where you move characters in all directions, or if you’ve marveled at the physics of roller coasters, you’ve seen vectors in action! Understanding how vectors interact can provide clarity to complex systems—just like solving a puzzle.

In Conclusion: Keep Exploring

You know, there’s something incredibly satisfying about working through mathematical challenges like these. Maybe it’s that "aha!" moment when everything clicks, or the confidence that builds as you grasp complex concepts. No matter your motivation, always feel free to ask questions and seek clarity!

As you navigate your mathematical journey, remember that every equation, every vector tells a story. They’re pieces of a larger picture, waiting for you to uncover their significance. Embrace the challenge, and who knows what vector mysteries you might solve next? Happy calculating!