Vector Algebra

In this chapter: scalars and vectors, types of vectors, addition and the triangle and parallelogram laws, multiplication by a scalar, components and direction cosines, position vectors and the section formula, the scalar (dot) product and projection, and the vector (cross) product and area.

Basic ideas

A vector is a directed line segment; AB has initial point A and terminal point B, and its length |AB| is the magnitude. Quantities with magnitude only (mass, time, temperature) are scalars.

Addition and scalar multiplication

Triangle law: place the tail of b at the head of a; the vector from the tail of a to the head of b is a + b. Parallelogram law: if a and b are adjacent sides of a parallelogram from the same point, the diagonal from that point is a + b. The two laws are equivalent.

Addition is commutative and associative, and in any triangle ABC, AB + BC + CA = 0. Multiplying by a scalar λ changes the magnitude to |λ| |a| and keeps the direction if λ > 0 or reverses it if λ < 0. Two non-zero vectors are collinear exactly when b = λa for some scalar λ.

Components and direction cosines

With unit vectors î, ĵ, k̂ along the axes, the position vector of P(x, y, z) is r = xî + yĵ + zk̂, and

l, m, n are the direction cosines: the cosines of the angles the vector makes with the positive x, y and z axes. Any numbers proportional to them (such as x, y, z themselves) are direction ratios. Vectors in component form are added, subtracted and scaled component by component, and two vectors are equal only if all three components match.

The vector joining P(x₁, y₁, z₁) to Q(x₂, y₂, z₂) is PQ = (x₂ − x₁)î + (y₂ − y₁)ĵ + (z₂ − z₁)k̂, that is, position vector of Q minus position vector of P.

Section formula

If P and Q have position vectors a and b, the point R dividing PQ in the ratio m : n has position vector

Scalar (dot) product

• The result is a scalar. The dot product is commutative and distributes over addition.
• a · b = 0 for non-zero vectors means they are perpendicular.
• î · î = ĵ · ĵ = k̂ · k̂ = 1 and î · ĵ = ĵ · k̂ = k̂ · î = 0. Also a · a = |a|².
• cos θ = (a · b)/(|a| |b|) gives the angle between two vectors.

Projection

The scalar projection of a on b is |a| cos θ = (a · b)/|b|. The projection vector is this length times the unit vector b̂.

Worked example: Find the projection of a = 2î + 3ĵ + 2k̂ on b = î + 2ĵ + k̂.
Solution: a · b = 2(1) + 3(2) + 2(1) = 10 and |b| = √(1 + 4 + 1) = √6. Projection = 10/√6 = 5√6/3.

Worked example: Find the angle between a = î + ĵ − k̂ and b = î − ĵ + k̂.
Solution: a · b = 1 − 1 − 1 = −1, and |a| = |b| = √3. cos θ = −1/3, so θ = cos⁻¹(−1/3), an obtuse angle.

Vector (cross) product

• The result is a vector perpendicular to the plane of a and b.
• It is not commutative: b × a = −(a × b).
• a × b = 0 for non-zero vectors means they are parallel. In particular a × a = 0.
• î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ (cyclic order), and the reverse order gives the negatives.
• |a × b| is the area of the parallelogram with adjacent sides a and b; half of it is the area of the triangle. If the diagonals are d₁ and d₂, the parallelogram's area is ½|d₁ × d₂|.

Worked example: Find the area of the triangle with vertices A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1).
Solution: AB = ĵ + 2k̂ and AC = î + 2ĵ. Their cross product is î(1·0 − 2·2) − ĵ(0·0 − 2·1) + k̂(0·2 − 1·1) = −4î + 2ĵ − k̂, with magnitude √(16 + 4 + 1) = √21. Area = √21/2 square units.

For JEE, one more product is useful: the scalar triple product [a b c] = a · (b × c), equal to the determinant of the three rows of components. Its absolute value is the volume of the parallelepiped on the three vectors, and it is zero exactly when the vectors are coplanar.

Common mistakes: (1) Writing b × a = a × b; the sign changes. (2) Mixing up the conditions: dot product zero means perpendicular, cross product zero means parallel. (3) Forgetting the minus sign on the ĵ term when expanding the cross product determinant. (4) Dividing by |a| instead of |b| in the projection of a on b. (5) Using direction ratios as direction cosines without dividing by the magnitude.

JEE and MHT‑CET focus

• Unit vectors, direction cosines and the condition l² + m² + n² = 1.
• Section formula and collinearity of three points using position vectors.
• Angle between vectors, perpendicularity conditions and projections.
• Cross product for areas of triangles and parallelograms, and for a vector perpendicular to two given vectors.
• Identities such as |a × b|² + (a · b)² = |a|²|b|², and the scalar triple product for coplanarity and volume.

Practice questions