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Chapter 10: Vector Algebra

A vector is a quantity that has both magnitude and direction , displacement, velocity, force, angular momentum. Many of the physical world's most important quantities are vectors, and getting the algebra right transforms otherwise messy geometric problems into clean coordinate computations.

This chapter develops the formal vector toolkit. We start with vectors as directed line segments, then represent them with components in three-dimensional space using the unit vectors i^,j^,k^\hat{i}, \hat{j}, \hat{k}. We define addition geometrically (triangle/parallelogram rule) and componentwise. We define two distinct products: the scalar (dot) product a⃗⋅b⃗\vec{a}\cdot\vec{b} , a single number , and the vector (cross) product a⃗×b⃗\vec{a}\times\vec{b} , a new vector perpendicular to both inputs. Each unlocks a different class of geometric questions.

The dot product gives the angle between two vectors and projects one vector onto another. The cross product gives the area of the parallelogram spanned by two vectors and constructs a normal direction. Together with the scalar triple product [a⃗  b⃗  c⃗]=a⃗⋅(b⃗×c⃗)[\vec{a}\;\vec{b}\;\vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c}) , which gives the signed volume of the parallelepiped on three vectors , they form a complete set of tools for three-dimensional geometry.

For board exams the chapter is calculation-heavy: dot products, cross products, magnitudes, angles, areas. For JEE the same tools become geometric machinery, used to prove collinearity, coplanarity, perpendicularity, and to derive equations of lines and planes (which is the heart of Chapter 11).

A theme worth absorbing early: vectors do not depend on a coordinate system. You can express a⃗\vec{a} in any orthonormal basis; the magnitude, direction, and inner-product relations are unchanged. This makes vector methods coordinate-free, and often cleaner than synthetic Euclidean geometry.

Prerequisites: Cartesian three-dimensional coordinates from Class XI; trigonometric identities (especially the law of cosines and sine).

What's inside

  1. Vectors, magnitude, components , the basics; position vectors; direction cosines.
  2. Vector addition and scalar multiplication , geometric and algebraic rules.
  3. Section formula and position vectors , points dividing line segments in given ratios.
  4. Scalar (dot) product , definition, formula, projections, angle between vectors.
  5. Vector (cross) product , definition, determinant formula, area of parallelogram.
  6. Scalar triple product , volume of parallelepiped, coplanarity test.
  7. Applications in geometry , proving perpendicularity, collinearity, coplanarity.

Key results / Formula card

QuantityFormula
Position vectorOP⃗=xi^+yj^+zk^\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}
Magnitude$
Unit vector$\hat{a} = \vec{a}/
Direction cosines$l = a_1/
Section formula (internal, ratio m:nm:n)r⃗=mb⃗+na⃗m+n\vec{r} = \dfrac{m\vec{b} + n\vec{a}}{m + n}
Dot product$\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 =
Angle$\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{
Projection of a⃗\vec{a} on b⃗\vec{b}$\dfrac{\vec{a}\cdot\vec{b}}{
Cross producta⃗×b⃗\vec{a}\times\vec{b}: vector of magnitude $
Cross product (determinant)∣i^j^k^a1a2a3b1b2b3∣\begin{vmatrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}
Area of parallelogram$
Area of triangle$\dfrac{1}{2}
Scalar triple product[a⃗  b⃗  c⃗]=a⃗⋅(b⃗×c⃗)=det⁡(a1a2a3b1b2b3c1c2c3)[\vec{a}\;\vec{b}\;\vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c}) = \det \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix}
Coplanar[a⃗  b⃗  c⃗]=0[\vec{a}\;\vec{b}\;\vec{c}] = 0

How to read this chapter

Vectors reward visualisation. Sketch every problem , even a rough 2D sketch of a 3D situation can clarify which vectors point where. Internalise the geometric meaning of each operation: dot product = projection × magnitude; cross product = perpendicular vector with area-equal magnitude; triple product = signed volume. The algebraic formulas are tools; the geometry is the soul.

Sub-topics

6 pages
01
Vectors, magnitude, and components
Vectors as directed line segments; representing them with i^,j^,k^\hat{i}, \hat{j}, \hat{k}; magnitude and direction cosines.
02
Vector addition and the section formula
Triangle rule, parallelogram rule, position vectors, and the section formula for dividing line segments.
03
Scalar (dot) product
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec a||\vec b|\cos\theta , the operation that measures alignment, finds angles, and computes projections.
04
Vector (cross) product
A new vector perpendicular to both inputs, with magnitude equal to the area of the spanned parallelogram.
05
Scalar triple product
[a⃗ b⃗ c⃗]=a⃗⋅(b⃗×c⃗)[\vec a\,\vec b\,\vec c] = \vec a \cdot (\vec b \times \vec c) , the signed volume of the parallelepiped and the test for coplanarity.
06
Applications in geometry
Using vector algebra to prove collinearity, perpendicularity, parallelism, and to compute distances and angles.

Practice quiz

Answer the questions; explanations appear after each.

Quiz
Chapter 10 : Mixed practice
12 questions · pick the best answer
Q1

Magnitude of 3i^−4j^3\hat i - 4\hat j is

Q2

i^⋅j^\hat i \cdot \hat j equals

Q3

i^×j^\hat i \times \hat j equals

Q4

If a⃗⋅b⃗=0\vec a \cdot \vec b = 0 (neither zero), then

Q5

a⃗×b⃗=0⃗\vec a \times \vec b = \vec 0 implies

Q6

Direction cosines of i^+j^+k^\hat i + \hat j + \hat k

Q7

Area of triangle with sides a⃗,b⃗\vec a, \vec b

Q8

Volume of tetrahedron with edges a⃗,b⃗,c⃗\vec a, \vec b, \vec c from one vertex

Q9

Three vectors a⃗,b⃗,c⃗\vec a, \vec b, \vec c are coplanar iff

Q10

Section formula (internal, ratio m:nm:n): p⃗=\vec p =

Q11

Projection of a⃗\vec a on b⃗\vec b

Q12

Centroid of triangle with vertices a⃗,b⃗,c⃗\vec a, \vec b, \vec c