Lecture Notes: Determinants and Vectors

Introduction

Purpose of Lecture: Explain the importance of determinants in calculating vectors, specifically focusing on cross products and scalar triple products.
Previous Knowledge: Assumes understanding of dot products and determinant calculations from the previous lecture.

Cross Product: A mathematical operation that can be calculated using determinants.
Vectors Involved: Consider two vectors with coordinates (a, b, c) and (x, y, z).

Matrix Formation:
- Top Row: Standard unit vectors i, j, k.
- Middle Row: Coordinates of Vector U (a, b, c).
- Bottom Row: Coordinates of Vector V (x, y, z).
Cross Product: Determinant of this matrix.

Vectors Given:
- Vector U: 2i - j + k
- Vector V: i - 4j + 2k
Matrix Setup:
- [\begin{vmatrix} i & j & k \ 2 & -1 & 1 \ 1 & -4 & 2 \end{vmatrix}]
Determinant Calculation:
- i-component: (-1 \times 2 - (-4) \times 1 = 2)
- j-component: (2 \times 2 - 1 \times 1 = -3)
- k-component: (2 \times (-4) - (-1) \times 1 = -7)
Result: Cross product = 2i - 3j - 7k
Benefit: Easier than using trigonometry to find the perpendicular vector and magnitude.

Vectors Given:
- Vector U: 3i - j + k
- Vector V: i + 4j - 2k
- Vector W: 3i - 2j + 5k
Matrix Setup:
- [\begin{vmatrix} 3 & -1 & 1 \ 1 & 4 & -2 \ 3 & -2 & 5 \end{vmatrix}]
Determinant Calculation:
- First term: 3 ((4 \times 5 - (-2) \times 3) = 16)
- Second term: -1 ((1 \times 5 - (-2) \times 3) = 11)
- Third term: 1 ((1 \times (-2) - 4 \times 3) = -14)
Result: Scalar triple product = 48 + 11 - 4 = 45
Benefit: Easier than calculating vector and dot products individually.

Calculating cross products and scalar triple products using determinants simplifies the process compared to traditional methods involving trigonometry and vector direction calculations.
Determinants serve as a powerful tool for vector calculations when provided with coordinates.