## 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 \(\vec { AB }\) or simply as \(\vec { a }\). Here, the point A from where the vector \(\vec { AB }\) 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 \(\vec { AB }\) or \(\vec { a }\) is called magnitude of \(\vec { AB }\) or \(\vec { a }\) and it is represented by |\(\vec { AB }\)| or |\(\vec { a }\)| or a.

Note: Since, the length is never negative, so the notation |\(\vec { a }\)|< 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 \(\vec { OP }\) or \(\vec { r }\) is called the position vector of the point P with respect to O. The magnitude of \(\vec { OP }\) or \(\vec { r }\) is given by

**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 \(\frac { l }{ a } =\frac { m }{ b } =\frac { n }{ c } =\vec { r }\) Note: l^{2} + m^{2} + n^{2} = 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 \(\vec { 0 }\). Note: Zero vector cannot be assigned a definite direction or it may be regarded as having any direction. The vectors \(\vec { AA }\) , \(\vec { BB }\) represent the zero vector.

**Unit vector:** A vector of unit length is called unit vector. The unit vector in the direction of \(\vec { a }\) is \(\hat { a } =\frac { \vec { a } }{ \left| \vec { a } \right| }\)

**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. \(\vec { a }\) and \(\vec { b }\) are collinear, when \(\vec { a } =\pm \lambda \vec { b }\) or \(\vec { \left| a \right| } =\lambda \vec { \left| b \right| }\)

**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 \(\vec { a }\) = \(\vec { b }\), then \(\vec { \left| a \right| } =\vec { \left| b \right| }\) 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 \(\vec { BA }\) is negative of the vector \(\vec { AB }\) and written as \(\vec { BA }\) = – \(\vec { AB }\).

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, \(\vec { AB }\) = Position vector of \(\vec { B }\) – Positron vector of \(\vec { A }\)

= \(\vec { OB }\) – \(\vec { OA }\) = \(\vec { b }\) – \(\vec { a }\)

**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 \(\vec { BC }\) + \(\vec { CA }\) = \(\vec { BA }\) Note: The vector sum of three sides of a triangle taken in order is \(\vec { 0 }\).

**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 \(\vec { OA }\) and \(\vec { OC }\) respectively, then we get

\(\vec { OA }\) + \(\vec { OC }\) = \(\vec { OB }\)

Note: Both laws of vector addition are equivalent to each other.

**Properties of vector addition**

**Commutative:** For vectors \(\vec { a }\) and \(\vec { b }\), we have \(\vec { a } +\vec { b } =\vec { b } +\vec { a }\)

**Associative:** For vectors \(\vec { a }\), \(\vec { b }\) and \(\vec { c }\), we have \(\vec { a } +\left( \vec { b } +\vec { c } \right) =\left( \vec { a } +\vec { b } \right) +\vec { c }\)

Note: The associative property of vector addition enables us to write the sum of three vectors \(\vec { a }\), \(\vec { b }\) and \(\vec { c }\) as \(\vec { a } +\vec { b } +\vec { c }\) without using brackets.

**Additive identity:** For any vector \(\vec { a }\), a zero vector \(\vec { 0 }\) is its additive identity as \(\vec { a } +\vec { 0 } =\vec { a }\)

**Additive inverse:** For a vector \(\vec { a }\), a negative vector of \(\vec { a }\) is its additive inverse as \(\vec { a } +\left( \vec { -a } \right) =\vec { 0 }\)

**Multiplication of a Vector by a Scalar:** Let \(\vec { a }\) be a given vector and λ be a scalar, then multiplication of vector \(\vec { a }\) by scalar λ, denoted as λ \(\vec { a }\), is also a vector, collinear to the vector \(\vec { a }\) whose magnitude is |λ| times that of vector \(\vec { a }\) and direction is same as \(\vec { a }\), if λ > 0, opposite of \(\vec { a }\), if λ < 0 and zero vector, if λ = 0.

Note: For any scalar λ, λ . \(\vec { 0 }\) = \(\vec { 0 }\).

**Properties of Scalar Multiplication:** For vectors \(\vec { a }\), \(\vec { b }\) and scalars p, q, we have

(i) p(\(\vec { a }\) + \(\vec { a }\)) = p \(\vec { a }\) + p \(\vec { a }\)

(ii) (p + q) \(\vec { a }\) = p \(\vec { a }\) + q \(\vec { a }\)

(iii) p(q \(\vec { a }\)) = (pq) \(\vec { a }\)

Note: To prove \(\vec { a }\) is parallel to \(\vec { b }\), we need to show that \(\vec { a }\) = λ \(\vec { a }\), where λ is a scalar.

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

**Two dimensions:** If a point P in a plane has coordinates (x, y), then \(\vec { OP } =x\hat { i } +y\hat { j } \), where \(\hat { i }\) and \(\hat { j }\) are unit vectors along OX and OY-axes, respectively.

Then, \(\left| \vec { OP } \right| =\sqrt { { x }^{ 2 }+{ y }^{ 2 } }\)

**Three dimensions:** If a point P in a plane has coordinates (x, y, z), then \(\vec { OP } =x\hat { i } +y\hat { j } +z\hat { k }\), where \(\hat { i }\), \(\hat { j }\) and \(\hat { k }\) are unit vectors along OX, OY and OZ-axes, respectively. Then, \(\left| \vec { OP } \right| =\sqrt { { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } }\)

**Vector Joining of Two Points:** If P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) are any two points, then the vector joining P_{1} and P_{2} is the vector \(\vec { { P }_{ 1 }{ P }_{ 2 } }\)

**Section Formula:** Position vector \(\vec { OR }\) of point R, which divides the line segment joining the points A and B with position vectors \(\vec { a }\) and \(\vec { b }\) respectively, internally in the ratio m : n is given by

For external division,

Note: Position vector of mid-point of the line segment joining end points A(\(\vec { a }\)) and B(\(\vec { b }\)) is given by \(\vec { OR } =\frac { \vec { a } +\vec { b } }{ 2 }\)

**Dot Product of Two Vectors:** If θ is the angle between two vectors \(\vec { a }\) and \(\vec { b }\), then the scalar or dot product denoted by \(\vec { a }\) . \(\vec { b }\) is given by \(\vec { a } \cdot \vec { b } =\left| \vec { a } \right| \left| \vec { b } \right| cos\theta\), where 0 ≤ θ ≤ π.

Note:

(i) \(\vec { a } \cdot \vec { b }\) is a real number

(ii) If either \(\vec { a } =\vec { 0 }\) or \(\vec { b } =\vec { 0 }\), then θ is not defined.

Properties of dot product of two vectors \(\vec { a }\) and \(\vec { b }\) are as follows:

**Vector (or Cross) Product of Vectors:** If θ is the angle between two non-zero, non-parallel vectors \(\vec { a }\) and \(\vec { b }\), then the cross product of vectors, denoted by \(\vec { a } \times \vec { b }\) is given by

where, \(\hat { n }\) is a unit vector perpendicular to both \(\vec { a }\) and \(\vec { b }\), such that \(\vec { a }\), \(\vec { b }\) and \(\hat { n }\) form a right handed system.

Note

(i) \(\vec { a } \times \vec { b }\) is a vector quantity, whose magnitude is \(\left| \vec { a } \times \hat { b } \right| =\left| \vec { a } \right| \vec { \left| b \right| } sin\theta\)

(ii) If either \(\vec { a } =\vec { 0 }\) or \(\vec { b } =\vec { 0 }\), then0is not defined.

Properties of cross product of two vectors \(\vec { a }\) and \(\vec { b }\) are as follows: