Vectors
Vectors
- A vector has size and direction: $\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$.
- magnitude $|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}$; a unit vector has magnitude 1.
- A line through $\mathbf{a}$ in direction $\mathbf{b}$: $\mathbf{r} = \mathbf{a} + t\mathbf{b}$. Lines may be parallel, intersect, or be skew.
Practice
What is the magnitude of the vector i + 2j + 2k?
|v| = √(1² + 2² + 2²) = √(1+4+4) = √9 = 3.
The scalar (dot) product
$$\mathbf{a}\cdot\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\mathbf{a}|\,|\mathbf{b}|\cos\theta$$
- So $\mathbf{a}\cdot\mathbf{b} = 0$ means the vectors are perpendicular.
- Use it to find the angle between two vectors.
Practice
Find the scalar product (i + 2j + 2k)·(2i + 2j + k).
(1)(2) + (2)(2) + (2)(1) = 2 + 4 + 2 = 8.
Practice
If a·b = 0, the two vectors are perpendicular.
cos θ = 0 gives θ = 90°, so a zero dot product means the vectors are perpendicular.
You've got it
Key idea
- magnitude $|\mathbf{v}| = \sqrt{x^2+y^2+z^2}$; line: $\mathbf{r} = \mathbf{a} + t\mathbf{b}$
- scalar product $\mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf a||\mathbf b|\cos\theta$
- $\mathbf{a}\cdot\mathbf{b} = 0$ → perpendicular