Maharashtra Board Class 12 Maths Part 1 Chapter 5 Vectors PDF Download

Vectors

Let's Study

1. Vectors and their types.

2. Section formula.

3. Dot Product of Vectors.

4. Cross Product of Vectors.

5. Triple Product of Vectors.

Let's Recall

Scalar quantity: A quantity which can be completely described by magnitude only is called a scalar quantity. For example, mass, length, temperature, area, volume, time, distance, speed, work, money, voltage, density, resistance, etc. In this book, scalars are given by real numbers.

Vector quantity: A quantity which needs to be described using both magnitude and direction is called a vector quantity. For example, displacement, velocity, force, electric field, acceleration, momentum, etc.

Let's Learn

5.1 Representation of Vector

Vector is represented by a directed line segment.

If AB is a segment and its direction is shown with an arrowhead, then the directed segment AB has magnitude as well as direction. This is an example of vector.

The segment AB with direction from A to B denotes the vector AB, read as AB bar, while direction from B to A denotes the vector BA.

In vector AB, the point A is called the initial point and the point B is called the terminal point.

The directed line segment is a part of a line of unlimited length which is called the line of support or the line of action of the given vector.

If the initial and terminal point are not specified then the vectors are denoted by a, b, c, or a, b, c (bold face) etc.

Teacher's Note

A vector shows both size and direction. Think of a ship going north - the distance is size, and north is the direction.

Exam Trick

Remember: A vector has two parts - magnitude (how much) and direction (which way). A scalar has only magnitude.

Points to Remember

A vector is shown as an arrow with a starting point and ending point.
The starting point is called initial point.
The ending point is called terminal point.
Vectors are written with a line or arrow on top like AB or a.
Two important types are scalar (size only) and vector (size and direction).

5.1.1 Magnitude of a Vector

The magnitude (or size or length) of AB is denoted by |AB| and is defined as the length of segment AB. That is, |AB| = l(AB).

Magnitudes of vectors a, b, c are |a|, |b|, |c| respectively.

The magnitude of a vector does not depend on its direction. Since the length is never negative, |a| ≥ 0.

5.1.2 Types of Vectors

i) Zero Vector: A vector whose initial and terminal points coincide is called a zero vector (or null vector) and denoted as 0. Zero vector cannot be assigned a definite direction and it has zero magnitude or it may be regarded as having any suitable direction. The vectors AA, BB represent the zero vector and |AA| = 0.

ii) Unit Vector: A vector whose magnitude is unity (that is, 1 unity) is called a unit vector. The unit vector in the direction of a given vector a is denoted by â, read as a-cap or a-hat.

iii) Co-initial and Co-terminus Vectors: Vectors having same initial point are called co-initial vectors, whereas vectors having same terminal point are called co-terminus vectors. Here a and b are co-initial vectors. c and d are co-terminus vectors.

iv) Equal Vectors: Two or more vectors are said to be equal vectors if they have same magnitude and direction.

As |a| = |b|, and their directions are same regardless of initial point, we write as a = b.

Here |a| = |c|, but directions are not same, so a ≠ c.

Here directions of a and d are same but |a| ≠ |d|, so a ≠ d.

v) Negative of a Vector: If a is a given vector, then the negative of a is a vector whose magnitude is same as that of a but whose direction is opposite to that of a. It is denoted by –a.

Thus, if PQ = a, then QP = –a = –PQ.

Here |PQ| = |QP|.

vi) Collinear Vectors: Vectors are said to be collinear vectors if they are parallel to same line or they are along the same line.

vii) Free Vectors: If a vector can be translated anywhere in the space without changing its magnitude and direction then such a vector is called free vector. In other words, the initial point of free vector can be taken anywhere in the space keeping magnitude and direction same.

Teacher's Note

Equal vectors look the same - same length and same direction. Just like two cars traveling in the same direction at the same speed.

Exam Trick

Remember: Zero vector = starting and ending point are the same. Unit vector = magnitude is exactly 1. Opposite vectors = same size, opposite direction.

Points to Remember

Zero vector has both starting and ending points at same location.
Unit vector always has magnitude equal to 1.
Equal vectors have same magnitude and same direction.
Negative vector has same size but opposite direction.
Collinear vectors lie on the same line or parallel lines.

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