What is 10.1 Introduction in NCERT Class 12 Mathematics Chapter 4?

10.1 Introduction is the topic of Part 1 of Chapter 4 (Ch 4 — Chapter 4) in NCERT Class 12 Mathematics. This lesson builds intuition through examples, visuals, and graded practice.

A vector is a quantity that has both magnitude and direction u2014 for example, displacement, velocity, force. It is represented geometrically by a directed line segment whose length gives the magnitude and whose arrow gives the direction.

A scalar is a quantity having only magnitude u2014 temperature, mass, time, density. Scalars obey ordinary arithmetic; vectors require their own algebra.

A vector is a quantity that has both magnitude and direction — for example, displacement, velocity, force. It is represented geometrically by a directed line segment whose length gives the magnitude and whose arrow gives the direction.

A scalar is a quantity having only magnitude — temperature, mass, time, density. Scalars obey ordinary arithmetic; vectors require their own algebra.

TOPIC 14 OF 25

Vector Algebra — Introduction, Types and Direction Cosines

Q: What are direction cosines?

For a vector r = ai + bj + ck, the direction cosines are l = a/|r|, m = b/|r|, n = c/|r| u2014 the cosines of the angles the vector makes with the positive x, y, z axes. They satisfy lu00b2 + mu00b2 + nu00b2 = 1.

Q: What is a unit vector?

A vector of magnitude 1. The unit vector along r is ru0302 = r/|r|. The standard unit vectors along the x, y, z axes are u00ee, u0135, ku0302.

Q: What is the difference between direction cosines and direction ratios?

Direction cosines are normalised to satisfy lu00b2 + mu00b2 + nu00b2 = 1 (unique up to sign). Direction ratios are any three numbers proportional to the direction cosines u2014 non-unique but easier to read off from the components a, b, c of the vector itself.

Q: What is a position vector?

The position vector of a point P with coordinates (x, y, z) is the vector OP = xi + yj + zk pointing from the origin O to P. Its magnitude |OP| = u221a(xu00b2 + yu00b2 + zu00b2) is the distance from origin.

🎓 Class 12 Mathematics CBSE Theory Ch 10 — Vector Algebra ⏱ ~15 min

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10.1 Introduction

In our everyday life we encounter many physical quantities. Some — like distance, mass, area, volume, temperature, speed — can be specified by a single number plus a unit (e.g. "10 km", "5 kg"). These are scalars^?.

Others require both a magnitude and a direction. "Walk 5 km" is incomplete — north? south? east? Vectors^? are the natural language for these directed quantities: displacement, velocity, acceleration, force, momentum, electric field, … . The compact algebra of vectors developed in this chapter is foundational for physics, engineering, computer graphics and machine learning.

William Rowan Hamilton

1805 – 1865

The Irish mathematician who, in 1843, invented quaternions — 4-dimensional numbers with non-commutative multiplication — and from them developed modern vector analysis. The terms "vector" and "scalar" were coined by Hamilton. His work was made practical for physics by Gibbs and Heaviside, who distilled the 3-D vector calculus we still teach today.

10.2 Some Basic Concepts

Vector — geometric representation

A vector is a directed line segment. We write \(\vec{AB}\) to mean the vector from \(A\) (tail) to \(B\) (head). The magnitude \(|\vec{AB}|\) is the length of the segment; the direction is given by the arrow.

We also use bold or arrowed lower-case letters: \(\mathbf{a}=\vec{a}\). Two vectors are equal when they have the same magnitude AND the same direction (regardless of position).

A vector \(\vec{a}=\vec{AB}\) has magnitude (length) and direction (arrow).

Position vector

The position vector of a point \(P(x,y,z)\) is \(\vec{OP}=x\,\hat\imath+y\,\hat\jmath+z\,\hat k\), where \(O\) is the origin and \(\hat\imath,\hat\jmath,\hat k\) are unit vectors along the positive x, y, z axes. Its magnitude: \[|\vec{OP}|=\sqrt{x^2+y^2+z^2}.\]

10.2.1 Direction cosines and direction ratios

Direction cosines and ratios

Let \(\vec{r}=a\hat\imath+b\hat\jmath+c\hat k\) make angles \(\alpha,\beta,\gamma\) with the positive x, y, z axes respectively. The cosines are \[\boxed{\;l=\cos\alpha=\dfrac{a}{|\vec r|},\ m=\cos\beta=\dfrac{b}{|\vec r|},\ n=\cos\gamma=\dfrac{c}{|\vec r|}\;}\] called the direction cosines. They satisfy \(l^2+m^2+n^2=1\).

The triple \((a, b, c)\) (or any non-zero scalar multiple) is called the direction ratios. Direction cosines = direction ratios divided by \(|\vec r|\).

10.3 Types of Vectors

Six standard types

Zero (null) vector \(\vec 0\): magnitude 0, no specific direction. Represented by a point.
Unit vector \(\hat a\): magnitude 1, denoted with a hat. Standard unit vectors: \(\hat\imath,\hat\jmath,\hat k\).
Equal vectors: same magnitude AND direction (location irrelevant).
Negative \(-\vec a\): same magnitude as \(\vec a\), opposite direction.
Parallel / collinear vectors: same or opposite direction (i.e. \(\vec b=\lambda\vec a\) for some scalar \(\lambda\)).
Coplanar vectors: three or more vectors lying in (or parallel to) the same plane.

Worked Examples

Example 1. Identify each as scalar or vector: (i) Time period, (ii) Distance, (iii) Force, (iv) Velocity, (v) Work done, (vi) Acceleration, (vii) Density, (viii) 30 km/hr.

Scalars: (i) time period, (ii) distance, (v) work done, (vii) density, (viii) speed (30 km/hr — speed, no direction).
Vectors: (iii) force, (iv) velocity, (vi) acceleration.

Example 2. Find the position vector of the point \(P(1, 2, 3)\) and its magnitude.

\(\vec{OP}=\hat\imath+2\hat\jmath+3\hat k\). \(|\vec{OP}|=\sqrt{1+4+9}=\sqrt{14}\).

Example 3. Find the direction cosines of \(\vec r=2\hat\imath-3\hat\jmath+\hat k\).

\(|\vec r|=\sqrt{4+9+1}=\sqrt{14}\). Direction cosines: \(l=2/\sqrt{14},\ m=-3/\sqrt{14},\ n=1/\sqrt{14}\). Check: \(l^2+m^2+n^2=4/14+9/14+1/14=14/14=1\). ✓

Example 4. Find a unit vector along \(\vec a=3\hat\imath+\hat\jmath-2\hat k\).

\(|\vec a|=\sqrt{9+1+4}=\sqrt{14}\). Unit vector: \(\hat a=\dfrac{1}{\sqrt{14}}(3\hat\imath+\hat\jmath-2\hat k)\). Note \(|\hat a|=1\). ✓

Example 5. If a vector makes equal angles with the three coordinate axes, find these angles.

Equal direction cosines \(l=m=n\) with \(l^2+m^2+n^2=1\): \(3l^2=1\Rightarrow l=\pm 1/\sqrt 3\). So \(\cos\alpha=\pm 1/\sqrt 3\), giving \(\alpha\approx 54.74°\) (or its supplement 125.26°).

Activity: Compute Direction Cosines

L3 Apply

Materials: Pen, paper, calculator.

Predict: A vector along \(\hat\imath+\hat\jmath+\hat k\) makes the same angle with each axis. Find this angle and verify using direction cosines.

\(\vec r=\hat\imath+\hat\jmath+\hat k\). \(|\vec r|=\sqrt 3\).
Direction cosines: \(l=m=n=1/\sqrt 3\).
Verify: \(l^2+m^2+n^2=3\cdot 1/3=1\). ✓
Angle: \(\cos^{-1}(1/\sqrt 3)\approx 54.74°\).
Now find direction cosines of \(2\hat\imath-\hat\jmath+2\hat k\). (\(|\vec r|=3\), so \(l=2/3, m=-1/3, n=2/3\); check sum of squares = 1.)

A vector along the diagonal of a unit cube has direction cosines \(1/\sqrt 3\) along each axis. The angle 54.74° appears in crystallography, molecular geometry (tetrahedral bond angle ≈ 109.47° = 2·54.74°), and antenna design.

Competency-Based Questions

Scenario: A drone flies from origin O to point P(4, 3, 12) in 3-D space. Treating its displacement as a vector \(\vec{OP}\).

Q1. Magnitude of the displacement vector:

L3 Apply

Answer: \(|\vec{OP}|=\sqrt{16+9+144}=\sqrt{169}=13\) units.

Q2. Direction cosines:

L3 Apply

Answer: \(l=4/13,\ m=3/13,\ n=12/13\). Check: 16/169+9/169+144/169=169/169=1 ✓.

Q3. (T/F) "If two vectors have the same magnitude, they are equal." Justify.

L5 Evaluate

False. Vector equality requires same magnitude AND same direction. Two velocity vectors of 5 m/s pointing east and west have equal magnitude but opposite direction; they are negatives of each other, not equal.

Q4. Identify scalar/vector: (a) age, (b) acceleration, (c) area, (d) electric field, (e) work, (f) momentum.

L2 Understand

Answer: Scalars: age, area, work. Vectors: acceleration, electric field, momentum. (Some texts treat area as a vector when oriented — but as a scalar in elementary geometry.)

Q5. Design: a 3-D position sensor reports a robot's coordinates as P(6, −2, 3). Find the unit vector pointing from origin to P, and the angle the displacement makes with the z-axis.

L6 Create

Solution: \(|\vec{OP}|=\sqrt{36+4+9}=\sqrt{49}=7\). Unit vector: \(\hat r=\dfrac{1}{7}(6\hat\imath-2\hat\jmath+3\hat k)\). \(\cos\gamma=3/7\), so \(\gamma=\cos^{-1}(3/7)\approx 64.6°\).

Assertion–Reason Questions

Assertion (A): The direction cosines of any non-zero vector satisfy \(l^2+m^2+n^2=1\).
Reason (R): The direction cosines are obtained by dividing the components by the magnitude.

(a) Both true, R explains A.

(b) Both true, R doesn't explain A.

(c) A true, R false.

(d) A false, R true.

Answer: (a). By construction, \(l^2+m^2+n^2=(a^2+b^2+c^2)/|\vec r|^2=1\). R is the construction.

Assertion (A): Two collinear vectors must be parallel.
Reason (R): Collinear means lying on the same line or parallel lines, which is the same as having the same or opposite direction.

(a) Both true, R explains A.

(b) Both true, R doesn't explain A.

(c) A true, R false.

(d) A false, R true.

Answer: (a). "Collinear" and "parallel" are synonymous in vector terminology.

Assertion (A): The zero vector \(\vec 0\) has no defined direction.
Reason (R): A vector of magnitude 0 is just a point; there is no arrow to specify a direction.

(a) Both true, R explains A.

(b) Both true, R doesn't explain A.

(c) A true, R false.

(d) A false, R true.

Answer: (a). Geometric reason matches the algebraic definition.

Frequently Asked Questions — Vector Algebra — Introduction, Types and Direction Cosines

What is a vector?

A vector is a quantity having both magnitude and direction. Represented as a directed line segment.

What is a scalar?

A scalar is a quantity with only magnitude — temperature, mass, time, density.

What are direction cosines?

l = a/|r|, m = b/|r|, n = c/|r| — cosines of the angles with x, y, z axes. Satisfy l² + m² + n² = 1.

What is a unit vector?

A vector of magnitude 1. Unit vector along r is r̂ = r/|r|.

What is the difference between direction cosines and direction ratios?

Direction cosines are unique (up to sign) and satisfy l²+m²+n²=1. Direction ratios are any 3 numbers proportional to the cosines.

What is a position vector?

The vector OP from the origin O to a point P(x, y, z) — i.e. xi + yj + zk.

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