Grade 11 Unit 5:5.3.3 Cross Product & Exercise 5.12 & Exercise 5 .13 | Saquama

Introduction

In this tutorial, we will explore the concept of the cross product in vectors, as discussed in the Grade 11 Maths curriculum. We'll break down the cross product, provide examples through Exercises 5.12 and 5.13, and highlight key points to aid your understanding. This guide is suitable for students looking to deepen their knowledge of vector mathematics.

Step 1: Understanding the Cross Product

The cross product of two vectors results in a third vector that is perpendicular to the plane formed by the original vectors. It is denoted as:

• If A and B are vectors, the cross product is written as A × B.

Key Characteristics of the Cross Product:

• Magnitude: The magnitude of the cross product can be calculated using the formula:

  [ |A × B| = |A| |B| \sin(θ) ]

  where θ is the angle between vectors A and B.

• Direction: The direction of the resulting vector is determined using the right-hand rule:

  • Point your index finger in the direction of A and your middle finger in the direction of B. Your thumb will point in the direction of A × B.

Practical Tip:

When calculating the cross product, ensure that your vectors are represented in component form (i.e., A = (Ax, Ay, Az) and B = (Bx, By, Bz)) for easier computation.

Step 2: Calculating the Cross Product

To calculate the cross product of vectors in component form, use the following determinant formula:

[ A × B = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} ]

Steps for Calculation:

1. Set up the determinant using the unit vectors i, j, and k.

2. Compute the determinant:

   • For i: ( (A_y \cdot B_z - A_z \cdot B_y) )
   • For j: ( (A_z \cdot B_x - A_x \cdot B_z) )
   • For k: ( (A_x \cdot B_y - A_y \cdot B_x) )

3. Combine results into the vector format:

   • A × B = (i, j, k)

Example:

Let A = (1, 2, 3) and B = (4, 5, 6).

1. Set up the determinant: [ A × B = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 2 & 3 \ 4 & 5 & 6 \end{vmatrix} ]

2. Calculate:

   • i: ( (2 \cdot 6 - 3 \cdot 5) = 12 - 15 = -3 )
   • j: ( (3 \cdot 4 - 1 \cdot 6) = 12 - 6 = 6 )
   • k: ( (1 \cdot 5 - 2 \cdot 4) = 5 - 8 = -3 )

3. Result: [ A × B = (-3, 6, -3) ]

Step 3: Exercises 5.12 and 5.13

Now, let's apply what we've learned through Exercises 5.12 and 5.13.

Exercise 5.12

• Problem: Find the cross product of vectors C = (2, 0, -1) and D = (1, 3, 0).
• Solution Steps:
  1. Set up the determinant as shown in Step 2.
  2. Compute the determinant for each component.
  3. Present the resultant vector.

Exercise 5.13

• Problem: Given vectors E = (0, -2, 1) and F = (3, 1, 2), find the cross product.
• Solution Steps:
  1. Set up the determinant.
  2. Calculate each component.
  3. Combine the results into the final vector.

Conclusion

In this tutorial, we explored the cross product of vectors, learned how to calculate it using the determinant method, and applied our knowledge through specific exercises. Understanding the cross product is crucial for various applications in physics and engineering. As a next step, practice additional problems involving vector calculations to reinforce your skills.