CBSE Class 12 Maths Notes Chapter 10 Vector Algebra
Vector: Those quantities which have magnitude, as well as direction, are called vector quantities or vectors.
Note: Those quantities which have only magnitude and no direction, are called scalar quantities.
Representation of Vector: A directed line segment has magnitude as well as direction, so it is called vector denoted as or simply as
Magnitude of a Vector: The length of the vector
Note: Since, the length is never negative, so the notation |
Position Vector: Let O(0, 0, 0) be the origin and P be a point in space having coordinates (x, y, z) with respect to the origin O. Then, the vector
Direction Cosines: If α, β and γ are the angles which a directed line segment OP makes with the positive directions of the coordinate axes OX, OY and OZ respectively, then cos α, cos β and cos γ are known as the direction cosines of OP and are generally denoted by the letters l, m and n respectively.
i.e. l = cos α, m = cos β, n = cos γ Let l, m and n be the direction cosines of a line and a, b and c be three numbers, such that
Types of Vectors
Null vector or zero vector: A vector, whose initial and terminal points coincide and magnitude is zero, is called a null vector and denoted as
Unit vector: A vector of unit length is called unit vector. The unit vector in the direction of
Collinear vectors: Two or more vectors are said to be collinear, if they are parallel to the same line, irrespective of their magnitudes and directions, e.g.
Coinitial vectors: Two or more vectors having the same initial point are called coinitial vectors.
Equal vectors: Two vectors are said to be equal, if they have equal magnitudes and same direction regardless of the position of their initial points. Note: If
Negative vector: Vector having the same magnitude but opposite in direction of the given vector, is called the negative vector e.g. Vector
Note: The vectors defined above are such that any of them may be subject to its parallel displacement without changing its magnitude and direction. Such vectors are called ‘free vectors’.
To Find a Vector when its Position Vectors of End Points are Given: Let a and b be the position vectors of end points A and B respectively of a line segment AB. Then,
=
Addition of Vectors
Triangle law of vector addition: If two vectors are represented along two sides of a triangle taken in order, then their resultant is represented by the third side taken in opposite direction, i.e. in ∆ABC, by triangle law of vector addition, we have
Parallelogram law of vector addition: If two vectors are represented along the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the sides. If the sides OA and OC of parallelogram OABC represent
Note: Both laws of vector addition are equivalent to each other.
Properties of vector addition
Commutative: For vectors
Associative: For vectors
Note: The associative property of vector addition enables us to write the sum of three vectors
Additive identity: For any vector
Additive inverse: For a vector
Multiplication of a Vector by a Scalar: Let
Note: For any scalar λ, λ .
Properties of Scalar Multiplication: For vectors
(i) p(
(ii) (p + q)
(iii) p(q
Note: To prove
Components of a Vector: Let the position vector of P with reference to O is
Two dimensions: If a point P in a plane has coordinates (x, y), then
Then,
Three dimensions: If a point P in a plane has coordinates (x, y, z), then
Vector Joining of Two Points: If P1(x1, y1, z1) and P2(x2, y2, z2) are any two points, then the vector joining P1 and P2 is the vector
Section Formula: Position vector
For external division,
Note: Position vector of mid-point of the line segment joining end points A(
Dot Product of Two Vectors: If θ is the angle between two vectors
Note:
(i)
(ii) If either
Properties of dot product of two vectors
Vector (or Cross) Product of Vectors: If θ is the angle between two non-zero, non-parallel vectors
where,
Note
(i)
(ii) If either
Properties of cross product of two vectors