Cross Product Of Vectors A And B: Calculation & Solution

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Alright, guys, let's dive into a classic vector problem! We've got two vectors, A and B, and we need to find their cross product. This operation is super useful in physics and engineering for things like calculating torque or angular momentum. So, grab your thinking caps, and let's get to it!

Understanding the Cross Product

Before we jump into the calculation, let's quickly recap what the cross product actually is. The cross product, denoted by A × B, results in a new vector that is perpendicular to both A and B. The magnitude of this new vector is equal to the area of the parallelogram formed by A and B. The direction is determined by the right-hand rule. If you point your fingers in the direction of A and curl them towards B, your thumb will point in the direction of A × B.

Why is this useful? Imagine you're trying to tighten a bolt with a wrench. The force you apply is a vector, and the distance from the bolt to where you're applying the force is another vector. The torque, which determines how effectively you're tightening the bolt, is the cross product of these two vectors! So, understanding cross products is essential for understanding how forces cause rotation.

Furthermore, the cross product isn't commutative, meaning A × B is not the same as B × A. In fact, A × B = - (B × A). This is because switching the order reverses the direction of the resulting vector. This anti-commutative property highlights the importance of the order in which you perform the cross product.

Setting up the Problem

We're given:

  • Vector A = -2i + 3j + k
  • Vector B = i + 4j + k

Our mission, should we choose to accept it (and we do!), is to find A × B.

Let's get down to business. Remember, the i, j, and k components represent the unit vectors along the x, y, and z axes, respectively. We can represent vectors A and B as ordered triples:

  • A = (-2, 3, 1)
  • B = (1, 4, 1)

Now, we're ready to roll!

Calculating the Cross Product

The cross product can be calculated using a determinant of a matrix. It looks a bit scary at first, but it's actually quite straightforward once you get the hang of it. Here's the setup:

A × B = | i  j  k |
        | -2 3  1 |
        | 1  4  1 |

To compute this determinant, we expand along the first row:

A × B = i * (determinant of the 2x2 matrix formed by removing the first row and first column) - j * (determinant of the 2x2 matrix formed by removing the first row and second column) + k * (determinant of the 2x2 matrix formed by removing the first row and third column)

Let's break it down piece by piece:

  • i component: The determinant of the 2x2 matrix | 3 1 | | 4 1 | is (3 * 1) - (1 * 4) = 3 - 4 = -1. So, the i component is -1i.
  • j component: The determinant of the 2x2 matrix | -2 1 | | 1 1 | is (-2 * 1) - (1 * 1) = -2 - 1 = -3. So, the j component is -(-3)j = 3j. Notice the minus sign in front of the j component! This is part of the formula.
  • k component: The determinant of the 2x2 matrix | -2 3 | | 1 4 | is (-2 * 4) - (3 * 1) = -8 - 3 = -11. So, the k component is -11k.

Putting it all together, we get:

A × B = -1i + 3j - 11k

The Answer and Why It Matters

Therefore, the cross product of vector A and vector B is -i + 3j - 11k. Looking back at the options, the correct answer is (E).

But wait, there's more! It's not enough to just crunch the numbers; it's crucial to understand what this result means. The vector -i + 3j - 11k is perpendicular to both A and B. If you were to visualize these three vectors in 3D space, you'd see that A × B points in a direction that's orthogonal to the plane formed by A and B.

Common Mistakes to Avoid

  • Forgetting the minus sign for the j component: This is a super common mistake! Always remember to subtract the determinant when calculating the j component.
  • Mixing up the order of subtraction in the 2x2 determinants: Make sure you're calculating (top-left * bottom-right) - (top-right * bottom-left).
  • Getting the signs wrong: Pay close attention to the signs of the components of the original vectors. A single sign error can throw off the entire calculation.
  • Not understanding the concept: Memorizing the formula is not enough. Make sure you understand why you are doing each step. This will help you catch errors and apply the concept to different situations.

Alternative Methods (For the Curious Minds)

While the determinant method is the most common, there are other ways to think about the cross product. You can also use the distributive property, remembering the cross products of the unit vectors:

  • i × i = 0
  • j × j = 0
  • k × k = 0
  • i × j = k
  • j × k = i
  • k × i = j
  • j × i = -k
  • k × j = -i
  • i × k = -j

Using this method, you'd expand (-2i + 3j + k) × (i + 4j + k) term by term and simplify. It's a bit more tedious, but it can be a good way to double-check your work.

Level Up Your Understanding

To really master cross products, I recommend practicing a variety of problems. Try changing the components of vectors A and B and see how the cross product changes. Also, try visualizing the vectors in 3D space to get a better feel for their orientation.

Challenge yourself! Try calculating the area of the parallelogram formed by vectors A and B. Remember that the magnitude of the cross product is equal to the area.

Conclusion

So, there you have it! We've successfully calculated the cross product of two vectors. Remember, the cross product is a powerful tool with applications in various fields. Keep practicing, and you'll become a cross product pro in no time! Keep up the great work, guys, and happy calculating!