Problem Solving using Vectors

Parallel Vectors: Two vectors $\vec{a}$ and $\vec{b}$ are parallel if and only if one is a scalar multiple of the other, such that $\vec{a} = k\vec{b}$ for some non-zero scalar $k$. If $k > 0$, the vectors point in the same direction; if $k < 0$, they point in opposite directions.

Collinear Points: Three points $A$, $B$, and $C$ are collinear (lie on the same straight line) if the vectors $\vec{AB}$ and $\vec{BC}$ are parallel and share a common point. To prove collinearity, one must demonstrate that $\vec{AB} = k\vec{BC}$ and explicitly state that the shared point $B$ ensures they lie on the same line rather than just being parallel segments.

Geometric Proofs: This principle is frequently used to prove that midpoints of shapes align or that specific intersections divide segments in predictable ways.

The Ratio Formula: If a point $P$ divides a line segment $AB$ in the ratio $\lambda : \mu$, the position vector $\vec{OP}$ can be found using the starting point and a fraction of the total displacement. The formula is expressed as $\vec{OP} = \vec{OA} + \frac{\lambda}{\lambda + \mu}\vec{AB}$.

Alternative Form: This can also be written in terms of position vectors $\vec{a}$ and $\vec{b}$ as $\vec{OP} = \frac{\mu}{\lambda + \mu}\vec{a} + \frac{\lambda}{\lambda + \mu}\vec{b}$. This weighted average approach is useful when the origin is the primary reference point.

Midpoints: In the specific case of a midpoint, the ratio is $1:1$, simplifying the position vector to $\vec{OM} = \frac{1}{2}(\vec{OA} + \vec{OB})$.

Parallelograms: In a parallelogram $ABCD$, opposite sides are represented by equal vectors, meaning $\vec{AB} = \vec{DC}$ and $\vec{AD} = \vec{BC}$. This property allows for the calculation of a missing vertex by setting up a vector journey from a known point.

Vector Journeys: To find an unknown vector, you can follow any path of known vectors. For example, $\vec{AC} = \vec{AB} + \vec{BC}$. The result is independent of the path taken, provided the start and end points are the same.

Centroids and Intersections: Vectors are used to find where medians or diagonals intersect by setting up two different vector paths to the same point and solving for the unknown scalars.

Finding Angles: The angle $\theta$ between two vectors can be found using the Cosine Rule within a triangle formed by the vectors. If vectors $\vec{a}$ and $\vec{b}$ form two sides of a triangle, the third side is $\vec{b} - \vec{a}$, and the angle is found via $\cos \theta = \frac{|a|^2 + |b|^2 - |b-a|^2}{2|a||b|}$.

Area of a Triangle: The area of a triangle defined by two vectors $\vec{a}$ and $\vec{b}$ originating from the same point is given by $\text{Area} = \frac{1}{2}|a||b|\sin \theta$. This requires first calculating the magnitudes (lengths) of the vectors and the angle between them.

Component Resolution: Vectors can be broken into horizontal and vertical components using $x = r \cos \theta$ and $y = r \sin \theta$, where $r$ is the magnitude and $\theta$ is the direction angle.