Vector Algebra & 3D Geometry
Syllabus Unit: Mathematics
Why it matters
Chapter overview
Vector algebra equips you to handle quantities with both magnitude and direction — from basic vector operations and the dot/cross products, through the vector and Cartesian equations of lines and planes in 3D, to direct applications like the section formula, work and torque — skills that also underpin the mechanics questions in the Physics portion of the GAT.
Mathematics & General Ability Mathematics 40% weightage
Live Animation1 / 4
1Identify componentsWrite the vector as xi+yj+zk
2Compute magnitude|a| = √(x²+y²+z²)
3Apply the triangle lawAdd vectors head-to-tail to find the resultant
4Normalize to a unit vectorâ = a/|a|, isolating pure direction
1Vectors & Basic Operations
A vector carries both magnitude and direction, and every vector can be broken into perpendicular components or combined by the triangle law.
Learning Objectives
- Distinguish a vector (magnitude + direction) from a scalar (magnitude only), with examples.
- Represent a vector in component form xi+yj+zk and compute its magnitude.
- Add and subtract vectors using the triangle law and parallelogram law.
- Find a unit vector in the direction of a given vector.
Key Terms
Position VectorThe vector from the origin O to a point P, denoted OP or r.
Unit Vector (â)A vector of magnitude 1 in the direction of a, computed as â = a/|a|.
Triangle Law of AdditionIf a and b are placed head to tail, the resultant a+b is the third side of the triangle they form.
Null (Zero) VectorA vector of zero magnitude and undefined direction, e.g. the result of a − a.
A **scalar** has only magnitude (e.g. mass, temperature); a **vector** has both magnitude and direction (e.g. force, velocity, displacement).
Component form and magnitude
a = xi + yj + zk, |a| = √(x²+y²+z²)
Adding vectors
- Triangle law — place vectors head-to-tail; the resultant closes the triangle.
- Parallelogram law — place vectors tail-to-tail; the resultant is the diagonal of the parallelogram they form.
Unit vector: â = a / |a|
Exam TipQuick check: if a claimed unit vector's components squared don't sum to exactly 1, it isn't actually a unit vector.
Common MistakeVector addition is NOT simple addition of magnitudes unless the vectors are collinear and point the same way — direction always matters.