Vectors – Dot and Cross Products
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Direct answer
The dot product a·b = |a||b| cos theta is a scalar that tests perpendicularity (a·b = 0) and gives projections; the cross product a × b is a vector of magnitude |a||b| sin theta, directed by the right-hand rule, testing parallelism (a × b = 0) and measuring areas. The scalar triple product a·(b × c) equals the determinant of the components, gives the volume of the parallelepiped and vanishes exactly when the three vectors are coplanar.
What you must remember
- Dot product: a·b = a1b1 + a2b2 + a3b3 = |a||b| cos theta; cos theta = (a·b)/(|a||b|); the projection of a on b is (a·b)/|b| — divide by the vector projected onto.
- Cross product: |a × b| = |a||b| sin theta; area of the parallelogram on a and b is |a × b| and of the triangle half of that; a unit vector perpendicular to both is (a × b)/|a × b|.
- Scalar triple product: [a b c] = a·(b × c) = the determinant of the component rows; volume of the parallelepiped is |[a b c]| and of the tetrahedron (1/6)|[a b c]|; zero means coplanar; swapping two vectors flips the sign, cycling preserves it.
- Vector triple product: a × (b × c) = (a·c)b - (a·b)c; vector multiplication is not associative, and this expansion is the only safe route through nested crosses.
- Centroid of a triangle with vertices a, b, c: (a + b + c)/3; section formula for internal ratio m : n: (n a + m b)/(m + n).
- Angle bisector: the internal bisector of two vectors points along a/|a| + b/|b| — the sum of their unit vectors.
- Distribution: a·(b + c) = a·b + a·c for both products — run arguments component-free until the end.
Common confusion
Scalar versus vector outputs. a·(b × c) is a number; a × (b × c) is a vector — writing one for the other destroys the question. Projection direction is the second slip: the projection of a on b divides by |b|, not |a|. And [a b c] is orientation-sensitive — a single row swap flips its sign, so keep a fixed cyclic order.
Exam-focused takeaway
JEE Main asks angles between vectors, areas of triangles and parallelograms, volumes via determinants and coplanarity checks — the fastest reliable marks in the paper when determinants are clean. JEE Advanced layers the triple products: proving identities such as a × (b × c) + b × (c × a) + c × (a × b) = 0, decomposing vectors along a basis, and configurations settled by scalar triple products. Component-free manipulation first, components only to finish.
Frequently asked questions
What does a·b = 0 tell you?
The vectors are perpendicular (or one is zero); contrast a × b = 0, which signals parallel vectors.
How do I find the area of a triangle given its vertices?
Take two side vectors from one vertex and halve the magnitude of their cross product; the parallelogram on the same vectors takes the full magnitude.
What is the volume of a tetrahedron in vector form?
(1/6) of the absolute scalar triple product of the three edge vectors from one vertex; the parallelepiped takes the full |[a b c]|.
Is vector multiplication associative?
No; a × (b × c) and (a × b) × c are generally different — expand through (a·c)b - (a·b)c instead of regrouping.
How do I test whether four points are coplanar?
Form three vectors from one point to the other three and set their scalar triple product to zero; non-zero means volume, hence non-coplanar.