MCV4U: Complete Vector Guide
1. Vector Fundamentals
1.1 Geometric vs Algebraic Vectors
Geometric: Directed line segment \( \overrightarrow{AB} \)
Algebraic: \( \mathbf{v} = \langle v_x, v_y, v_z \rangle \)
From \( A(1,2) \) to \( B(4,6) \): \( \mathbf{v} = \langle 3,4 \rangle \)
2. Vector Operations
2.1 Addition/Subtraction
Component-wise operations: \[ \mathbf{u} \pm \mathbf{v} = \langle u_x \pm v_x, u_y \pm v_y, u_z \pm v_z \rangle \]
2.2 Scalar Multiplication
\( k\mathbf{v} = \langle kv_x, kv_y, kv_z \rangle \)
3. Key Vector Concepts
3.1 Unit Vectors
\( \hat{v} = \frac{\mathbf{v}}{|\mathbf{v}|} \) (Direction vector)
Standard basis: \( \hat{i} = \langle 1,0,0 \rangle \), \( \hat{j} \), \( \hat{k} \)
3.2 Magnitude
\( |\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \)
4. Dot Product
\( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta = u_xv_x + u_yv_y + u_zv_z \)
Application: Angle Between Vectors
\( \cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|} \)
5. Cross Product
\( \mathbf{u} \times \mathbf{v} = |\mathbf{u}||\mathbf{v}|\sin\theta\ \hat{n} \)
Determinant formula with \( \hat{i}, \hat{j}, \hat{k} \) components
Right-Hand Rule Application
Used to determine direction of resulting vector
6. Vector Projections
6.1 Scalar Projection
\( \text{comp}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|} \)
6.2 Vector Projection
\( \text{proj}_{\mathbf{v}}\mathbf{u} = \left(\frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2}\right)\mathbf{v} \)
7. Real-World Applications
- Force analysis in physics
- Computer graphics transformations
- Engineering stress calculations
8. MCV4U Practice Problems
Problem 1: Cross Product
Find \( \mathbf{a} \times \mathbf{b} \) where \( \mathbf{a} = \langle 2, -1, 3 \rangle \) and \( \mathbf{b} = \langle 4, 0, -2 \rangle \)
Solution
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 3 \\ 4 & 0 & -2 \end{vmatrix} = \langle 2, 16, 4 \rangle \]
9. Key Takeaways
- Vectors have magnitude and direction
- Cross product results in a perpendicular vector
- Dot product helps find angles between vectors
- Projections essential for force decomposition