Questions
4 questions per board paper
Difficulty
Medium
Importance
Core — never skip
Overview
Vector Algebra deals with quantities having both magnitude and direction, serving as a fundamental mathematical tool for engineering physics and calculus. Mastering this topic is essential for CBSE boards and entrance exams, as it provides the basis for coordinate geometry and 3D space analysis.
Types of Vectors
Vectors are defined by their magnitude and direction in space, often represented in component form. Understanding the distinct properties of unit, null, and position vectors is essential for building complex vector equations.
- Position Vector OP = xi + yj + zk
- Magnitude |a| = sqrt(x^2 + y^2 + z^2)
- Unit Vector a^ = a / |a|
- Direction Cosines: cos(alpha) = x/|a|, cos(beta) = y/|a|, cos(gamma) = z/|a|
- Collinear vectors satisfy a = kb
Scalar (Dot) Product
The scalar product yields a real number and is primarily used to determine the angle between two vectors or the projection of one vector onto another. It is commutative and follows distributive laws over addition.
- a . b = |a||b|cos(theta)
- a . b = x1x2 + y1y2 + z1z2
- Perpendicular condition: a . b = 0
- Projection of a on b = (a . b) / |b|
- i.i = j.j = k.k = 1
- i.j = j.k = k.i = 0
Vector (Cross) Product
The vector product results in a vector perpendicular to the plane containing the two original vectors. It is non-commutative and is the go-to method for calculating areas of triangles and parallelograms.
- a x b = |a||b|sin(theta)n^
- a x b = Determinant of [i, j, k; x1, y1, z1; x2, y2, z2]
- Parallel condition: a x b = 0
- Area of Triangle = 0.5 * |a x b|
- Area of Parallelogram = |a x b|
- i x i = j x j = k x k = 0
Formula Sheet
|a| = sqrt(x^2 + y^2 + z^2)
cos(theta) = (a . b) / (|a||b|)
a . b = x1x2 + y1y2 + z1z2
|a x b| = |a||b|sin(theta)
Area = 0.5 * |AB x AC|
Unit vector a^ = a / |a|
cos^2(alpha) + cos^2(beta) + cos^2(gamma) = 1
Exam Tip
Always verify if the question asks for a scalar projection or a vector projection, as the latter requires an additional multiplication by the unit vector of the direction line.
Common Mistakes
- Confusing the dot product (scalar) with the cross product (vector) result.
- Forgetting to multiply the unit vector n^ in the cross product magnitude formula.
- Errors in calculating the determinant when performing cross products with negative components.
More Revision Notes
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