Vector Algebra
This domain tests vector addition, section formula, scalar and vector products, projection, and areas of triangles and parallelograms. Questions are usually 1–3 marks but appear in almost every paper.
Core rules
- Unit vector along a is â = a/|a|.
- a·b = |a||b| cos θ; a·b = 0 means perpendicular.
- a×b = |a||b| sin θ n̂; a×b = 0 means parallel. Note b×a = −(a×b).
- Projection of a on b = (a·b)/|b|.
- |a×b| is the area of the parallelogram with sides a and b; half of it is the area of the triangle.
- Scalar triple product [a b c] = a·(b×c) gives the volume of the parallelepiped; zero means coplanar.
Worked example
Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5), C(1, 5, 5). AB = i + 2j + 3k and AC = 4j + 3k. Then AB × AC = (−6)i + (−3)j + 4k, whose magnitude is √(36 + 9 + 16) = √61. Area = ½|AB × AC| = √61/2 square units.
Traps
- Reversing the order in a cross product and losing the sign.
- Using |a×b| when the question asks for the triangle but forgetting the factor ½.
- Confusing projection with the component vector.
- Forgetting to convert given points into direction vectors before crossing.