Vector Algebra

Vector algebra forms the backbone of analytical geometry and physics in JEE Mathematics. A vector is a quantity possessing both magnitude and direction, represented as a = a₁i + a₂j + a₃k in three-dimensional space, where i, j, k are unit vectors along the coordinate axes.

The magnitude of vector a is |a| = $\sqrt{a₁² + a₂² + a₃²}$. A unit vector in the direction of a is a/|a|. Two vectors are equal if and only if their corresponding components are identical. The zero vector 0 has magnitude zero and arbitrary direction.

Addition follows the parallelogram law or triangle law: a + b = (a₁+b₁)i + (a₂+b₂)j + (a₃+b₃)k. Scalar multiplication scales each component: ka = ka₁i + ka₂j + ka₃k.

The dot product (scalar product) is defined as a . b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos($\theta$), where $\theta$ is the angle between the vectors. Key properties: commutative, distributive over addition, a . a = |a|². Two non-zero vectors are perpendicular if and only if a . b = 0.

The cross product (vector product) is a x b = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k, computed via the determinant of the 3x3 matrix with i, j, k in the first row. Its magnitude |a x b| = |a||b|sin($\theta$) equals the area of the parallelogram formed by a and b. The direction follows the right-hand rule. Key properties: anti-commutative (a x b = -b x a), distributive, and a x a = 0.

The projection of b onto a is (a . b/|a|²)a, and the scalar projection (component) is a . b/|a|.

Section formula: A point dividing the line joining points with position vectors a and b in ratio m:n is (mb + na)/(m+n) for internal division and (mb - na)/(m-n) for external division.

Linear dependence: Vectors a, b, c are linearly dependent if there exist scalars x, y, z (not all zero) such that xa + yb + zc = 0. Two vectors are collinear if one is a scalar multiple of the other. Three vectors are coplanar if their scalar triple product vanishes.

The key problem-solving concept is recognizing when to apply dot product (for angles, projections, perpendicularity) versus cross product (for areas, perpendicular vectors, and testing parallelism), and using the component form systematically.

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