Vectors in 3 Dimensions

Three-Dimensional Representation: A 3D vector describes a movement or position in space relative to an origin $(0, 0, 0)$, incorporating $x$, $y$, and $z$ components. These are represented either as column vectors $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$ or in unit vector form as $x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$.

Unit Vector Basis: The unit vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ represent a displacement of 1 unit in the positive direction of the $x$, $y$, and $z$ axes respectively. This standard basis allows any vector to be decomposed into independent orthogonal movements, simplifying complex spatial arithmetic.

Position Vectors: A position vector $\vec{OA}$ specifically denotes the displacement of point $A$ from the origin $O$. This is distinct from a displacement vector $\vec{AB}$, which represents the relative movement required to travel from point $A$ to point $B$ in 3D space.

Extension of Pythagoras' Theorem: The magnitude of a 3D vector $\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ is derived by applying Pythagoras' theorem twice. First, the diagonal in the $xy$-plane is calculated, and then that result is combined with the $z$ height to find the spatial diagonal.

Distance Between Points: To find the distance between points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, one calculates the magnitude of the displacement vector $\vec{AB}$. This involves squaring the differences of each coordinate component, summing them, and taking the square root.

Scalar Linearity: Scalar multiplication of a 3D vector scales every component proportionally. Geometrically, this results in a vector that is either longer, shorter, or reversed in direction but remains parallel to the original vector line.

Unit Vector Construction: A unit vector in the direction of $\mathbf{a}$ is created by dividing the vector by its own magnitude, expressed as $\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$. This normalization is essential when only the direction is required for a calculation, such as in force projections.

Vector Arithmetic: Addition and subtraction are performed component-wise, meaning $x$ values are added to $x$ values, $y$ to $y$, and $z$ to $z$. This follows the 'triangle law' or 'parallelogram law' of vector addition, where the result represents the cumulative displacement of the individual vectors.

Parallelism Verification: Two vectors are parallel if one can be expressed as a scalar multiple of the other, $\mathbf{a} = \lambda \mathbf{b}$. To check this in 3D, verify if the ratio between the $x, y,$ and $z$ components of the two vectors is constant.

Negative Component Squaring: When a component is negative, such as $-3\mathbf{j}$, ensure that the negative sign is included inside the square when calculating magnitude: $(-3)^2 = 9$. Errors often occur when students calculate $-3^2 = -9$, leading to incorrect magnitudes.

Misidentifying Parallel Vectors: Vectors are only parallel if the ratio of ALL components is the same. For example, $\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \\ 5 \end{pmatrix}$ are NOT parallel because the $z$-ratio ($1/1.2$) differs from the $x$ and $y$ ratios ($1/2$).

Magnitude vs Squared Magnitude: Students often confuse the square of the distance with the distance itself. Always read the question carefully to see if an 'exact' distance (likely a surd) or a decimal approximation is required.