Revise Vectors for Cambridge 9709 (3.7)

In 3D space, a vector can be written in component form using the standard unit vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, which represent movement along the x, y, and z axes respectively. For example, the vector $\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$ can also be written as a column vector. A position vector is a special type of vector that starts at the origin (0, 0, 0) and ends at a specific point in space.

$\mathbf{v} = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}$

The magnitude (or modulus) of a vector is its length. It's calculated using Pythagoras' theorem extended to three dimensions.

For a vector $\mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$, its magnitude is $|\mathbf{a}| = \sqrt{x^2 + y^2 + z^2}$.

The scalar product is one way to 'multiply' two vectors. The result is a scalar (a number), not another vector, which is why it's called the scalar product. It has two definitions which we can equate to solve problems, most commonly to find the angle between two vectors.

Component form: For $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$, the scalar product is $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$.

Geometric form: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$, where $\theta$ is the angle between the vectors when placed tail-to-tail.

A straight line in 3D space can be uniquely defined if we know one point on the line and the direction in which the line travels. The vector equation of a line combines this information. The equation $\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$ describes the position vector $\mathbf{r}$ of any point on the line. It starts at a known point with position vector $\mathbf{a}$ and moves along the direction vector $\mathbf{d}$ by some scalar multiple $\lambda$.

$\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$\nWhere:\n$\mathbf{r}$ is the position vector of a general point on the line.\n$\mathbf{a}$ is the position vector of a specific, known point on the line.\n$\mathbf{d}$ is a direction vector parallel to the line.\n$\lambda$ is a scalar parameter.