Chapter 10 Vector Algebra is one of the most important chapters of Class 12 Mathematics. This chapter introduces vectors, which are quantities having both magnitude and direction. Vector algebra is widely used in physics, engineering, mechanics, and higher mathematics.
This chapter builds the foundation for Chapter 11 (Three-Dimensional Geometry), so understanding vectors clearly is very important. At Edu Tehri, this chapter is explained with easy language, diagrams, formulas, solved examples, handwritten notes, and free PDFs.
• Magnitude (size or value)
• Direction
• Displacement
• Velocity
• Force
• Distance
• Speed
• Mass
Scalars have no direction, but vectors always have direction.
• A directed line segment
• An arrow pointing in the direction of the vector
If a vector starts from point A and ends at point B, it is written as:
\vec {AB}
The length of the vector represents its magnitude.
|\vec{a}|
\vec{a} = ai + bj + ck
|\vec{a}| = \sqrt {a^2 + b^2 + c^2}
This formula is very important for exams.
A vector with zero magnitude is called a zero vector.
\vec{0}
Unit vector in direction of \vec{a}:
\hat {a} = \frac {\vec {a}} {|\vec {a}|}
• Same magnitude
• Same direction
A vector having same magnitude but opposite direction.
Vectors lying along the same or parallel line.
• cosα, cosβ, cosγ are called direction cosines
\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1
Direction ratios are proportional values of direction cosines.
A vector can be resolved into components along x, y, and z axes.
\vec{a} = ai + bj + ck
• a, b, c are components
• i, j, k are unit vectors along x, y, z axes
If two vectors are represented by two sides of a triangle taken in order, then their resultant is represented by the third side.
If two vectors act at a point and form adjacent sides of a parallelogram, then the diagonal through the point gives the resultant vector.
\vec{a} + \vec{b} = (a_1 + b_1)i + (a_2 + b_2)j + (a_3 + b_3)k
\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos \theta
Where θ is the angle between the vectors.
1. \vec{a} \cdot \vec{b} is a scalar
2. \vec{a} \cdot \vec{a} = |\vec{a}|^2
3. If vectors are perpendicular, dot product = 0
\vec{a} = a_1i + a_2j + a_3k
\vec{b} = b_1i + b_2j + b_3k
\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3
• Angle between two vectors
• Projection of one vector on another
• Whether vectors are perpendicular
\cos \theta = \frac {\vec{a} \cdot \vec{b}} {|\vec{a}||\vec{b}|}
\vec{a} \times \vec{b} = |\vec{a} || \vec{b} | \sin \theta \, \hat{n}
• θ = angle between vectors
• \hat{n} = unit vector perpendicular to both vectors
1. Result is a vector
2. Direction is given by right-hand thumb rule
3. If vectors are parallel, cross product = 0
\vec{a} \times \vec{b} =
\begin{vmatrix}
i & j & k \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3
\end{vmatrix}
This form is very important for exams.
• Area of parallelogram
• Area of triangle
• Vector perpendicular to a plane
\text{Area} = \frac{1}{2} | \vec{a} \times \vec{b}|
NCERT Chapter 10 contains 4 exercises.
• Exercise 10.1 – Basic vector concepts
• Exercise 10.2 – Magnitude and direction
• Exercise 10.3 – Dot product
• Exercise 10.4 – Cross product
• Usually 5–8 marks in board exams
• Often asked as long answer questions
• Forms base for 3D Geometry
• Free NCERT solution PDFs
• Handwritten notes
• Formula sheets
• Board-oriented questions
• Step-by-step explanations
• Concept-based chapter
• Highly scoring
• Important for competitive exams
• Visual and formula-oriented
• Foundation for next chapter
Because it is used in physics, engineering, and forms the base of 3D Geometry.
Yes, NCERT questions are fully sufficient for board exams.
Dot product and cross product are most important.
By learning formulas, practicing NCERT exercises, and avoiding calculation mistakes.