Vector Algebra

part 1

Basics of Vectors

• Scalars and vectors
• Types of vectors — unit, equal, collinear, coplanar
• Addition and subtraction of vectors
• Triangle and parallelogram law

part 2

Multiplication of Vectors

• Multiplication of vector by scalar
• Scalar (dot) product of two vectors
• Projection of vector on a line
• Vector (cross) product of two vectors

part 3

Applications of Vectors

• Scalar triple product
• Area of parallelogram and triangle using vectors
• Position vectors and section formula
• Applications in geometry

Dot (scalar) product

$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1 b_1+a_2 b_2+a_3 b_3$

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Angle between two vectors

$\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$

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Perpendicular vectors

$\vec{a} \perp \vec{b} \Leftrightarrow \vec{a}\cdot\vec{b} = 0$

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Cross (vector) product magnitude

$|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

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Cross product (determinant form)

$\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

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Magnitude of vector

$|\vec{a}| = \sqrt{a_1^2+a_2^2+a_3^2}$

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Unit vector

$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

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Section formula (internal division m:n)

$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$

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Parallel vectors

$\vec{a} \parallel \vec{b} \Leftrightarrow \vec{a}\times\vec{b} = \vec{0}$

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Area of parallelogram

$|\vec{a}\times\vec{b}|$

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Show that the vectors a⃗ = 2î − 3ĵ + 4k̂, b⃗ = î + 2ĵ − k̂ and c⃗ = 3î − ĵ + 2k̂ are coplanar by evaluating their scalar triple product.

Find the projection of the vector a⃗ = 2î + 3ĵ + 2k̂ on the vector b⃗ = î + 2ĵ + k̂.

If vectors a⃗ = 2î + 3ĵ + k̂ and b⃗ = î − 2ĵ + 4k̂, find |a⃗ × b⃗|. Also find a unit vector perpendicular to both a⃗ and b⃗.

If a⃗, b⃗, c⃗ are three vectors such that |a⃗| = 3, |b⃗| = 4, |c⃗| = 5 and each is perpendicular to the sum of the other two, find |a⃗ + b⃗ + c⃗|.

Find the angle between vectors a⃗ = î + ĵ + k̂ and b⃗ = î − ĵ + k̂.

Find the area of the triangle whose vertices are A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) using the cross product of vectors.

Find a unit vector in the direction of the vector a⃗ = 2î − 3ĵ + 6k̂.

If |a⃗| = 5, |b⃗| = 13 and |a⃗ × b⃗| = 25, find a⃗ · b⃗.

Write the value of λ for which the vectors a⃗ = 2î + λĵ + k̂ and b⃗ = 4î − 2ĵ + 2k̂ are parallel.

The position vectors of points A and B are î − ĵ + 2k̂ and 3î + 2ĵ − k̂. Find the position vector of the point P which divides AB in the ratio 2:1 (i) internally and (ii) externally.

How many marks is Vector Algebra worth in the Class 12 board exam?

Around 6 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Cross product and scalar triple product for area/volume.

Is Vector Algebra easy or hard?

It's rated medium — a moderate-difficulty chapter in Class 12 Maths. Most students need about 4 hours to cover it well.

What are the important topics in Vector Algebra?

The chapter covers 11 NCERT topics in 3 areas: Basics of Vectors; Multiplication of Vectors; Applications of Vectors.

What questions come from Vector Algebra in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Scalar Triple Product, Projection, Cross Product, Section Formula & Products — 1- to 5-mark questions.