Concept 01 / 06Vector Algebra
Vectors: Basic Concepts, Addition, Position Vectors and the Section Formula
Vectors as directed quantities, types of vectors (zero, unit, equal, collinear, coplanar), the triangle and parallelogram laws, properties of addition and scalar multiplication, position vectors, the section formula and vector proofs in geometry.
A vector is a quantity with both magnitude and direction, represented by a directed line segment from its initial point to its terminal point . Its magnitude is the length of the segment. Two vectors are equal if they have the same magnitude and direction, wherever they are drawn; vectors in this sense are 'free' and can be slid parallel to themselves.
Types of vectors. The zero vector has zero magnitude and no definite direction. A unit vector has magnitude ; the unit vector along is . Collinear (parallel) vectors lie along the same or parallel lines, regardless of sense; and are collinear iff for some scalar . Coplanar vectors lie in (or parallel to) one plane. Any two vectors are coplanar; three vectors may or may not be. The negative has the same magnitude and the opposite direction.
Addition. Triangle law: to add to , place the tail of at the head of ; the sum goes from the tail of to the head of . Parallelogram law: place both at a common tail; the sum is the diagonal from that point, as in the figure. The other diagonal is (from the head of to the head of ). Addition is commutative and associative, is the identity, and . For a closed polygon traversed in order, the sum of the side vectors is .
Triangle inequality: , with equality iff the vectors point the same way. Also .
Scalar multiplication. has magnitude and the direction of if , opposite if . It distributes over vector and scalar addition.
Position vectors. Fix an origin . The position vector of a point is , often written . Then for any two points ('head minus tail'). This single identity turns geometric statements into algebra.
Section formula. The point dividing internally in the ratio has position vector ; externally, . The midpoint is and the centroid of a triangle is .
Collinearity of points. Three points are collinear iff and are parallel, iff there are scalars , not all zero, with and . Similarly, four points are coplanar iff there is such a relation with four scalars summing to zero.
Vector methods in geometry. Many classical theorems become one-line calculations. The line joining the midpoints of two sides of a triangle is parallel to the third side and half as long: . The diagonals of a parallelogram bisect each other: both midpoints equal when .
Key Takeaways:
- ; .
- Parallelogram diagonals: and .
- (head minus tail).
- Section formula: ; centroid .
- Collinear points: with , not all zero.
- Sum of vectors around a closed polygon is .
- Pitfall: writing (tail minus head).
Brain-Teaser 1: is a regular hexagon with centre . Show that .
Solution: Pair the vectors symmetric about the diagonal . In a regular hexagon, and are rectangles, and this makes and each equal to . Indeed (opposite sides of the rectangle ), so ; similarly , so . Adding the remaining gives . Symmetric pairing plus the triangle law does the whole job without coordinates.
Brain-Teaser 2: Show that the points with position vectors , and are collinear, where are non-coplanar.
Solution: . . Since , the points are collinear, with the midpoint of . Non-coplanarity of is what lets us compare coefficients safely.
Brain-Teaser 3: Prove that the medians of a triangle are concurrent, and find the point.
Solution: Let the vertices be . The median from goes to . The point dividing it in the ratio is . This expression is symmetric in , so the same point lies on all three medians: they are concurrent at the centroid . Symmetry of the formula replaces the need to intersect lines.
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