NCERT Solutions Class 12 Maths Chapter 10 helps you master Vector Algebra through detailed solutions covering vector operations, dot and cross products, and scalar triple products. You’ll learn how to represent vectors in 3D space, calculate magnitudes and directions, prove vector identities, and solve problems involving position vectors and collinearity. These concepts are crucial for physics applications and form the foundation for 3D geometry in competitive exams.
Download Complete Chapter 10 Solutions PDF
All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra – Complete Guide
NCERT Class 12 Chapter 10 – Vector Algebra introduces you to one of the most powerful mathematical tools used in physics, engineering, and higher mathematics. You’ll explore vectors as quantities that have both magnitude and direction, learning how they differ from scalar quantities. This chapter carries 7 marks in your CBSE board exam, making it a high-importance topic that demands thorough understanding and consistent practice.
📊 CBSE Class 12 Maths Chapter 10 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 7 Marks |
| Unit Name | Vectors and 3D |
| Difficulty Level | Medium |
| Importance | High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
You will learn to represent vectors in different forms – geometrically, using directed line segments, and algebraically, using components. The chapter covers fundamental operations including vector addition using the triangle and parallelogram laws, scalar multiplication, and finding position vectors. You’ll understand unit vectors, direction cosines, and direction ratios, which form the foundation for 3D coordinate geometry.
The chapter then progresses to advanced concepts like the scalar (dot) product and vector (cross) product of two vectors. You’ll learn their geometric interpretations, properties, and applications in finding angles between vectors, projections, areas of parallelograms and triangles, and determining perpendicularity or parallelism of vectors. These concepts are frequently tested through 4-mark and 6-mark long answer questions in CBSE exams.
Quick Facts – Class 12 Chapter 10
| 📖 Chapter Number | Chapter 10 |
| 📚 Chapter Name | Vector Algebra |
| ✏️ Total Exercises | 2 Exercises |
| ❓ Total Questions | 24 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering Vector Algebra is essential not only for scoring well in your board exams but also for competitive examinations like JEE and NEET. The chapter connects beautifully with Three Dimensional Geometry and has practical applications in physics topics like mechanics and electromagnetism. With focused practice of NCERT solutions and previous year questions, you’ll develop strong problem-solving skills and confidence to tackle any vector-based question in your examination.
NCERT Solutions Class 12 Maths Chapter 10 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 10.1 | Complete step-by-step solutions for 5 questions | 📥 Download PDF |
| Exercise 10.2 | Complete step-by-step solutions for 19 questions | 📥 Download PDF |
Vector Algebra – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Position Vector \(\\vec{r} = x\\hat{i} + y\\hat{j} + z\\hat{k}\\) | Represents the position of a point (x, y, z) in space with respect to the origin. Note: Remember \\(\\hat{i}, \\hat{j}, \\hat{k}\\) are unit vectors along the x, y, and z axes respectively. | Finding the vector representing a point given its coordinates, or vice-versa. |
| Magnitude of a Vector \(|\\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\\) | Calculates the length or magnitude of a vector \\(\\vec{a} = a_1\\hat{i} + a_2\\hat{j} + a_3\\hat{k}\\). Note: Magnitude is always a non-negative scalar. | Finding the size of a vector, normalizing a vector (finding a unit vector). |
| Unit Vector \(\\hat{a} = \frac{\\vec{a}}{|\\vec{a}|}\\) | Finds a vector of length 1 in the same direction as the given vector \\(\\vec{a}\\). Note: Divide the vector by its magnitude. Make sure the magnitude is not zero. | Finding the direction of a vector, representing a direction with a vector of unit length. |
| Direction Cosines \(l = \frac{a_1}{|\\vec{a}|}, m = \frac{a_2}{|\\vec{a}|}, n = \frac{a_3}{|\\vec{a}|}\\) | Finds the cosines of the angles that a vector \\(\\vec{a} = a_1\\hat{i} + a_2\\hat{j} + a_3\\hat{k}\\) makes with the x, y, and z axes respectively. Note: l, m, and n are the direction cosines. Remember that \(l^2 + m^2 + n^2 = 1\). | Describing the orientation of a vector in space, finding the angles the vector makes with the coordinate axes. |
| Section Formula (Internal Division) \(\\vec{r} = \frac{m\\vec{b} + n\\vec{a}}{m + n}\\) | Finds the position vector of a point that divides the line segment joining points with position vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) internally in the ratio m:n. Note: Make sure you know which ratio corresponds to which vector. The point is BETWEEN A and B. | Finding the position of a point dividing a line segment in a given ratio (internal division). |
| Section Formula (External Division) \(\\vec{r} = \frac{m\\vec{b} – n\\vec{a}}{m – n}\\) | Finds the position vector of a point that divides the line segment joining points with position vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) externally in the ratio m:n. Note: The point is on the line AB, but OUTSIDE of the segment AB. | Finding the position of a point dividing a line segment in a given ratio (external division). |
| Dot Product \(\\vec{a} \\cdot \\vec{b} = |\\vec{a}| |\\vec{b}| \\cos{\\theta} = a_1b_1 + a_2b_2 + a_3b_3\\) | Calculates the dot product (scalar product) of two vectors. Note: If \\(\\vec{a} \\cdot \\vec{b} = 0\\), then \\(\\vec{a}\\) and \\(\\vec{b}\\) are perpendicular (assuming they are non-zero vectors). | Finding the angle between two vectors, checking if two vectors are perpendicular, finding the projection of one vector onto another. |
| Angle Between Two Vectors (using Dot Product) \(\\cos{\\theta} = \frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}| |\\vec{b}|}\\) | Calculates the angle \\(\\theta\\) between two vectors \\(\\vec{a}\\) and \\(\\vec{b}\\). Note: Remember to take the inverse cosine (arccos) to find the angle \\(\\theta\\). | Directly finding the angle between two vectors when their components or magnitudes are known. |
| Cross Product \(\\vec{a} \\times \\vec{b} = |\\vec{a}| |\\vec{b}| \\sin{\\theta} \\hat{n}\\) | Calculates the cross product (vector product) of two vectors. Note: The cross product results in a vector. \\(\\hat{n}\\) is the unit vector perpendicular to both \\(\\vec{a}\\) and \\(\\vec{b}\\). | Finding a vector perpendicular to two given vectors, calculating the area of a parallelogram or triangle formed by two vectors. |
| Area of Parallelogram \(Area = |\\vec{a} \\times \\vec{b}|\\) | Calculates the area of a parallelogram with adjacent sides represented by vectors \\(\\vec{a}\\) and \\(\\vec{b}\\). Note: Take the magnitude of the cross product. | Finding the area of a parallelogram when its sides are given as vectors. |
| Area of Triangle \(Area = \frac{1}{2} |\\vec{a} \\times \\vec{b}|\\) | Calculates the area of a triangle with two sides represented by vectors \\(\\vec{a}\\) and \\(\\vec{b}\\). Note: Take half the magnitude of the cross product. | Finding the area of a triangle when two of its sides are given as vectors. |
| Condition for Collinearity \(\\vec{a} \\times \\vec{b} = \\vec{0}\\) | Two vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) are collinear if their cross product is the zero vector. Note: This is equivalent to \\(\\vec{a} = k\\vec{b}\\) for some scalar k. | Proving that two vectors are parallel or lie on the same line. |
| Scalar Triple Product \([\\vec{a} \\vec{b} \\vec{c}] = \\vec{a} \\cdot (\\vec{b} \\times \\vec{c})\\) | The scalar triple product of three vectors. Note: Also equal to the determinant of the matrix formed by the components of the vectors. \[ [\\vec{a} \\vec{b} \\vec{c}] = \\begin{vmatrix} a_1 & a_2 & a_3 \\\\ b_1 & b_2 & b_3 \\\\ c_1 & c_2 & c_3 \\end{vmatrix} \] | Finding the volume of a parallelepiped, checking if three vectors are coplanar. |
| Volume of Parallelepiped \(V = |\\vec{a} \\cdot (\\vec{b} \\times \\vec{c})|\\) | Calculates the volume of a parallelepiped with adjacent edges represented by vectors \\(\\vec{a}, \\vec{b}, \\vec{c}\\). Note: Take the absolute value of the scalar triple product. | Finding the volume of a parallelepiped when its edges are given as vectors. |
| Coplanarity Condition \([\\vec{a} \\vec{b} \\vec{c}] = 0\\) | Three vectors \\(\\vec{a}, \\vec{b}, \\vec{c}\\) are coplanar if their scalar triple product is zero. Note: This means the volume of the parallelepiped formed by them is zero. | Proving that three vectors lie in the same plane. |
📚 Related Study Materials – Class 12 Maths Resources
| Resource | Access |
|---|---|
| NCERT Class 12 Sociology Textbook | Download Book |
| NCERT Class 10 Maths (Foundation) | View Solutions |