Vectors in 3 Dimensions

• A 3D vector represents a quantity with both magnitude and direction in a three-dimensional Cartesian coordinate system, typically defined by $x$, $y$, and $z$ axes.

• Vectors are expressed in unit vector form as $a = xi + yj + zk$, where $i$, $j$, and $k$ are perpendicular unit vectors of length 1 pointing along the positive $x$, $y$, and $z$ axes respectively.

• Alternatively, they can be written as column vectors $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$, which provides a clear vertical alignment for performing component-wise arithmetic operations.

• The position vector of a point $P(x, y, z)$ is the vector $\vec{OP}$ that starts at the origin $(0, 0, 0)$ and ends at point $P$.

• The magnitude (or modulus) of a 3D vector is calculated using a three-dimensional extension of Pythagoras' Theorem: $|a| = \sqrt{x^2 + y^2 + z^2}$.

• This principle works because any 3D vector can be viewed as the hypotenuse of a right-angled triangle where one side is the projection on the $xy$-plane and the other is the $z$-height.

• The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is found by calculating the magnitude of the vector connecting them: $d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$.

• Scalar multiplication scales the magnitude of a vector by a factor $k$ while maintaining or reversing its direction, affecting every component equally: $k(xi + yj + zk) = (kx)i + (ky)j + (kz)k$.

Vector Arithmetic

• Addition and Subtraction: Combine vectors by adding or subtracting their corresponding $i, j, k$ components independently. This represents the resultant movement of following one vector after another.
• Finding Unit Vectors: To find a unit vector $\hat{a}$ in the same direction as vector $a$, divide the vector by its own magnitude: $\hat{a} = \frac{a}{|a|}$. This is essential when only the direction of a force or movement is required without its specific scale.

Proving Geometric Properties

• Parallelism: Two vectors $a$ and $b$ are parallel if one is a scalar multiple of the other ($a = \lambda b$). If they also share a common point, the points are collinear (lying on the same straight line).
• Angle Calculation: The angle $\theta$ between a vector and the coordinate axes can be found using direction cosines: $\cos \theta_x = \frac{x}{|a|}$, $\cos \theta_y = \frac{y}{|a|}$, and $\cos \theta_z = \frac{z}{|a|}$.

• It is vital to distinguish between a position vector (relative to the origin) and a displacement vector (the path between two arbitrary points).

• While 2D vectors are always coplanar, 3D vectors allow for 'skew' lines that are neither parallel nor intersecting.

• Component Confusion: A common error is mixing up the $i, j, k$ order when converting between column and unit vector forms; always verify that $i$ corresponds to $x$, $j$ to $y$, and $k$ to $z$.

• Magnitude vs. Vector: Students often confuse the vector itself with its magnitude. Remember that a vector is a direction/movement, while magnitude is a scalar (a single number representing size).

• Origin Assumptions: Do not assume a vector starts at the origin unless it is explicitly stated as a position vector. Displacement vectors $\vec{AB}$ are calculated as $\vec{OB} - \vec{OA}$.