Parallel, Intersecting & Skew Lines

Parallel Condition: Two lines are parallel if their direction vectors are scalar multiples, such that $\mathbf{d_1} = k\mathbf{d_2}$ for some non-zero scalar $k$.

Identical Lines: If two lines are parallel and share at least one point, they are the same line; this is verified by checking if the position vector of a point on one line satisfies the equation of the other.

Vector Multiplicity: Unlike Cartesian equations, vector equations of the same line can look entirely different if they use different starting points or scaled direction vectors.

Equating General Points: To find an intersection, set the vector equations equal to each other using distinct parameters, typically $s$ and $t$, resulting in a system of three linear equations (one for each dimension $x, y, z$).

Simultaneous Solution: Solve any two of the three component equations to find potential values for $s$ and $t$ that would allow the lines to meet in those two dimensions.

The Consistency Check: Substitute the calculated $s$ and $t$ into the third (unused) equation; if the equation holds true, the lines intersect at a unique point.

Directional Angle: The angle $\theta$ between two lines is exclusively determined by the scalar product of their direction vectors $\mathbf{d_1}$ and $\mathbf{d_2}$ using the formula $\cos \theta = \frac{|\mathbf{d_1} \cdot \mathbf{d_2}|}{|\mathbf{d_1}| |\mathbf{d_2}|}$.

Perpendicularity Requirement: Two lines are perpendicular if and only if the scalar product of their direction vectors is zero, meaning $\mathbf{d_1} \cdot \mathbf{d_2} = 0$.

Scalar Result: The scalar product $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$ provides a numeric value that links vector algebra to trigonometric orientation.

The Foot of the Perpendicular: The shortest distance from a point $P$ to a line $l$ occurs at a point $F$ on the line such that the vector $\vec{PF}$ is perpendicular to the line's direction vector.

Optimization Method: By expressing the coordinates of $F$ in terms of the line's parameter $t$ and setting $(\vec{p} - \vec{f}) \cdot \mathbf{d} = 0$, one can solve for the specific $t$ that identifies the closest point.

Geometric Interpretation: The magnitude of the resulting vector $\vec{PF}$ represents the perpendicular distance, which is the absolute minimum distance between the external point and the line.

Parameter Distinction: When comparing two lines, always assign different letters to their scalars (e.g., $\lambda$ and $\mu$) to avoid the logical error of assuming the lines must reach a point at the same 'time' or parameter value.

Sanity Checks: Always verify that the intersection point calculated from $s$ and $t$ actually lies on both lines by re-inserting the values into the original vector equations.

Direction vs Position: In exam scenarios, ensure you are using the direction vector (the part multiplied by the scalar) for angle and parallelism tests, not the position vector.