Vector Algebra Class 12 Notes Maths Chapter 10

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 Image may be NSFW.
Clik here to view. $\vec { AB }$ or simply as Image may be NSFW.
Clik here to view. $\vec { a }$ . Here, the point A from where the vector Image may be NSFW.
Clik here to view. $\vec { AB }$ starts is called its initial point and the point B where it ends is called its terminal point.

Magnitude of a Vector: The length of the vector Image may be NSFW.
Clik here to view. $\vec { AB }$ or Image may be NSFW.
Clik here to view. $\vec { a }$ is called magnitude of Image may be NSFW.
Clik here to view. $\vec { AB }$ or Image may be NSFW.
Clik here to view. $\vec { a }$ and it is represented by |Image may be NSFW.
Clik here to view. $\vec { AB }$ | or |Image may be NSFW.
Clik here to view. $\vec { a }$ | or a.
Note: Since, the length is never negative, so the notation |Image may be NSFW.
Clik here to view. $\vec { a }$ |< 0 has no meaning.

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 Image may be NSFW.
Clik here to view. $\vec { OP }$ or Image may be NSFW.
Clik here to view. $\vec { r }$ is called the position vector of the point P with respect to O. The magnitude of Image may be NSFW.
Clik here to view. $\vec { OP }$ or Image may be NSFW.
Clik here to view. $\vec { r }$ is given by
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

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.
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

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 Image may be NSFW.
Clik here to view. $\frac { l }{ a } =\frac { m }{ b } =\frac { n }{ c } =\vec { r }$ Note: l² + m² + n² = 1

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 Image may be NSFW.
Clik here to view. $\vec { 0 }$ . Note: Zero vector cannot be assigned a definite direction or it may be regarded as having any direction. The vectors Image may be NSFW.
Clik here to view. $\vec { AA }$ , Image may be NSFW.
Clik here to view. $\vec { BB }$ represent the zero vector.

Unit vector: A vector of unit length is called unit vector. The unit vector in the direction of Image may be NSFW.
Clik here to view. $\vec { a }$ is Image may be NSFW.
Clik here to view. $\hat { a } =\frac { \vec { a } }{ \left| \vec { a } \right| }$

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. Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ are collinear, when Image may be NSFW.
Clik here to view. $\vec { a } =\pm \lambda \vec { b }$ or Image may be NSFW.
Clik here to view. $\vec { \left| a \right| } =\lambda \vec { \left| b \right| }$

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 Image may be NSFW.
Clik here to view. $\vec { a }$ = Image may be NSFW.
Clik here to view. $\vec { b }$ , then Image may be NSFW.
Clik here to view. $\vec { \left| a \right| } =\vec { \left| b \right| }$ but converse may not be true.

Negative vector: Vector having the same magnitude but opposite in direction of the given vector, is called the negative vector e.g. Vector Image may be NSFW.
Clik here to view. $\vec { BA }$ is negative of the vector Image may be NSFW.
Clik here to view. $\vec { AB }$ and written as Image may be NSFW.
Clik here to view. $\vec { BA }$ = – Image may be NSFW.
Clik here to view. $\vec { AB }$ .
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, Image may be NSFW.
Clik here to view. $\vec { AB }$ = Position vector of Image may be NSFW.
Clik here to view. $\vec { B }$ – Positron vector of Image may be NSFW.
Clik here to view. $\vec { A }$
= Image may be NSFW.
Clik here to view. $\vec { OB }$ – Image may be NSFW.
Clik here to view. $\vec { OA }$ = Image may be NSFW.
Clik here to view. $\vec { b }$ – Image may be NSFW.
Clik here to view. $\vec { a }$

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 Image may be NSFW.
Clik here to view. $\vec { BC }$ + Image may be NSFW.
Clik here to view. $\vec { CA }$ = Image may be NSFW.
Clik here to view. $\vec { BA }$ Note: The vector sum of three sides of a triangle taken in order is Image may be NSFW.
Clik here to view. $\vec { 0 }$ .
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

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 Image may be NSFW.
Clik here to view. $\vec { OA }$ and Image may be NSFW.
Clik here to view. $\vec { OC }$ respectively, then we get
Image may be NSFW.
Clik here to view. $\vec { OA }$ + Image may be NSFW.
Clik here to view. $\vec { OC }$ = Image may be NSFW.
Clik here to view. $\vec { OB }$
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10
Note: Both laws of vector addition are equivalent to each other.

Properties of vector addition
Commutative: For vectors Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ , we have Image may be NSFW.
Clik here to view. $\vec { a } +\vec { b } =\vec { b } +\vec { a }$

Associative: For vectors Image may be NSFW.
Clik here to view. $\vec { a }$ , Image may be NSFW.
Clik here to view. $\vec { b }$ and Image may be NSFW.
Clik here to view. $\vec { c }$ , we have Image may be NSFW.
Clik here to view. $\vec { a } +\left( \vec { b } +\vec { c } \right) =\left( \vec { a } +\vec { b } \right) +\vec { c }$
Note: The associative property of vector addition enables us to write the sum of three vectors Image may be NSFW.
Clik here to view. $\vec { a }$ , Image may be NSFW.
Clik here to view. $\vec { b }$ and Image may be NSFW.
Clik here to view. $\vec { c }$ as Image may be NSFW.
Clik here to view. $\vec { a } +\vec { b } +\vec { c }$ without using brackets.

Additive identity: For any vector Image may be NSFW.
Clik here to view. $\vec { a }$ , a zero vector Image may be NSFW.
Clik here to view. $\vec { 0 }$ is its additive identity as Image may be NSFW.
Clik here to view. $\vec { a } +\vec { 0 } =\vec { a }$

Additive inverse: For a vector Image may be NSFW.
Clik here to view. $\vec { a }$ , a negative vector of Image may be NSFW.
Clik here to view. $\vec { a }$ is its additive inverse as Image may be NSFW.
Clik here to view. $\vec { a } +\left( \vec { -a } \right) =\vec { 0 }$

Multiplication of a Vector by a Scalar: Let Image may be NSFW.
Clik here to view. $\vec { a }$ be a given vector and λ be a scalar, then multiplication of vector Image may be NSFW.
Clik here to view. $\vec { a }$ by scalar λ, denoted as λ Image may be NSFW.
Clik here to view. $\vec { a }$ , is also a vector, collinear to the vector Image may be NSFW.
Clik here to view. $\vec { a }$ whose magnitude is |λ| times that of vector Image may be NSFW.
Clik here to view. $\vec { a }$ and direction is same as Image may be NSFW.
Clik here to view. $\vec { a }$ , if λ > 0, opposite of Image may be NSFW.
Clik here to view. $\vec { a }$ , if λ < 0 and zero vector, if λ = 0.
Note: For any scalar λ, λ . Image may be NSFW.
Clik here to view. $\vec { 0 }$ = Image may be NSFW.
Clik here to view. $\vec { 0 }$ .

Properties of Scalar Multiplication: For vectors Image may be NSFW.
Clik here to view. $\vec { a }$ , Image may be NSFW.
Clik here to view. $\vec { b }$ and scalars p, q, we have
(i) p(Image may be NSFW.
Clik here to view. $\vec { a }$ + Image may be NSFW.
Clik here to view. $\vec { a }$ ) = p Image may be NSFW.
Clik here to view. $\vec { a }$ + p Image may be NSFW.
Clik here to view. $\vec { a }$
(ii) (p + q) Image may be NSFW.
Clik here to view. $\vec { a }$ = p Image may be NSFW.
Clik here to view. $\vec { a }$ + q Image may be NSFW.
Clik here to view. $\vec { a }$
(iii) p(q Image may be NSFW.
Clik here to view. $\vec { a }$ ) = (pq) Image may be NSFW.
Clik here to view. $\vec { a }$
Note: To prove Image may be NSFW.
Clik here to view. $\vec { a }$ is parallel to Image may be NSFW.
Clik here to view. $\vec { b }$ , we need to show that Image may be NSFW.
Clik here to view. $\vec { a }$ = λ Image may be NSFW.
Clik here to view. $\vec { a }$ , where λ is a scalar.

Components of a Vector: Let the position vector of P with reference to O is Image may be NSFW.
Clik here to view. $\vec { OP } =\vec { r } =x\hat { i } +y\hat { j } +z\hat { k }$ , this form of any vector is-called its component form. Here, x, y and z are called the scalar components of Image may be NSFW.
Clik here to view. $\vec { r }$ and Image may be NSFW.
Clik here to view. $x\hat { i }$ , Image may be NSFW.
Clik here to view. $y\hat { j }$ and Image may be NSFW.
Clik here to view. $z\hat { k }$ are called the vector components of Image may be NSFW.
Clik here to view. $\vec { r }$ along the respective axes.

Two dimensions: If a point P in a plane has coordinates (x, y), then Image may be NSFW.
Clik here to view. $\vec { OP } =x\hat { i } +y\hat { j }$ , where Image may be NSFW.
Clik here to view. $\hat { i }$ and Image may be NSFW.
Clik here to view. $\hat { j }$ are unit vectors along OX and OY-axes, respectively.
Then, Image may be NSFW.
Clik here to view. $\left| \vec { OP } \right| =\sqrt { { x }^{ 2 }+{ y }^{ 2 } }$
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

Three dimensions: If a point P in a plane has coordinates (x, y, z), then Image may be NSFW.
Clik here to view. $\vec { OP } =x\hat { i } +y\hat { j } +z\hat { k }$ , where Image may be NSFW.
Clik here to view. $\hat { i }$ , Image may be NSFW.
Clik here to view. $\hat { j }$ and Image may be NSFW.
Clik here to view. $\hat { k }$ are unit vectors along OX, OY and OZ-axes, respectively. Then, Image may be NSFW.
Clik here to view. $\left| \vec { OP } \right| =\sqrt { { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } }$
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

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 Image may be NSFW.
Clik here to view. $\vec { { P }_{ 1 }{ P }_{ 2 } }$
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

Section Formula: Position vector Image may be NSFW.
Clik here to view. $\vec { OR }$ of point R, which divides the line segment joining the points A and B with position vectors Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ respectively, internally in the ratio m : n is given by
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

For external division,
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

Note: Position vector of mid-point of the line segment joining end points A(Image may be NSFW.
Clik here to view. $\vec { a }$ ) and B(Image may be NSFW.
Clik here to view. $\vec { b }$ ) is given by Image may be NSFW.
Clik here to view. $\vec { OR } =\frac { \vec { a } +\vec { b } }{ 2 }$

Dot Product of Two Vectors: If θ is the angle between two vectors Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ , then the scalar or dot product denoted by Image may be NSFW.
Clik here to view. $\vec { a }$ . Image may be NSFW.
Clik here to view. $\vec { b }$ is given by Image may be NSFW.
Clik here to view. $\vec { a } \cdot \vec { b } =\left| \vec { a } \right| \left| \vec { b } \right| cos\theta$ , where 0 ≤ θ ≤ π.
Note:
(i) Image may be NSFW.
Clik here to view. $\vec { a } \cdot \vec { b }$ is a real number
(ii) If either Image may be NSFW.
Clik here to view. $\vec { a } =\vec { 0 }$ or Image may be NSFW.
Clik here to view. $\vec { b } =\vec { 0 }$ , then θ is not defined.

Properties of dot product of two vectors Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ are as follows:
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10
Image may be NSFW.
Clik here to view.

Vector (or Cross) Product of Vectors: If θ is the angle between two non-zero, non-parallel vectors Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ , then the cross product of vectors, denoted by Image may be NSFW.
Clik here to view. $\vec { a } \times \vec { b }$ is given by
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10

where, Image may be NSFW.
Clik here to view. $\hat { n }$ is a unit vector perpendicular to both Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ , such that Image may be NSFW.
Clik here to view. $\vec { a }$ , Image may be NSFW.
Clik here to view. $\vec { b }$ and Image may be NSFW.
Clik here to view. $\hat { n }$ form a right handed system.
Note
(i) Image may be NSFW.
Clik here to view. $\vec { a } \times \vec { b }$ is a vector quantity, whose magnitude is Image may be NSFW.
Clik here to view. $\left| \vec { a } \times \hat { b } \right| =\left| \vec { a } \right| \vec { \left| b \right| } sin\theta$
(ii) If either Image may be NSFW.
Clik here to view. $\vec { a } =\vec { 0 }$ or Image may be NSFW.
Clik here to view. $\vec { b } =\vec { 0 }$ , then0is not defined.

Properties of cross product of two vectors Image may be NSFW.
Clik here to view. $\vec { a }$ and Image may be NSFW.
Clik here to view. $\vec { b }$ are as follows:
Image may be NSFW.
Clik here to view. Vector Algebra Class 12 Notes Maths Chapter 10
Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.