Vectors

Navigating in 3D

• GPS satellites track your position in three dimensions. Aircraft navigate using heading, pitch, and roll.
• Vectors 向量 are the mathematics of direction and magnitude 模长 in space — essential for physics, engineering, and computer graphics.

Vectors in 3D

• 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 单位向量 has magnitude 1.

Forces like wind and water are vectors, with size and direction

Vector equations of lines

• A line through point $\mathbf{a}$ in direction $\mathbf{b}$: $\mathbf{r} = \mathbf{a} + t\mathbf{b}$.
• Lines may be parallel 平行 (same direction), intersect 相交 (share a point), or be skew 异面 (neither).

The scalar (dot) product

• So $\mathbf{a}\cdot\mathbf{b} = 0$ means the vectors are perpendicular 垂直.
• Use it to find the angle between two vectors: $\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$.

The scalar product 数量积 gives the angle: a·b = |a||b|cos θ, so cos θ = (a·b)/(|a||b|).

Worked example — angle between vectors

• $\mathbf{a} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$, $\mathbf{b} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k}$.
• $\mathbf{a}\cdot\mathbf{b} = 2 + 2 - 4 = 0$. So $\mathbf{a}$ and $\mathbf{b}$ are perpendicular ($\theta = 90^{\circ}$).

allow_no_figure: Vectors are inherently 3D and difficult to represent clearly in a 2D static diagram — the algebraic notation and worked examples convey the concepts effectively.

• Use position vectors and displacement vectors, and unit vectors; the ratio theorem divides a line in a given ratio.

You've got it

Write from a magnitude of 13

• The vector is 5 i plus 12 j.
• Find its magnitude.