Vectors Algebra | Chapter 10

Vectors Algebra | Chapter 10 | Class XII

Note: Click here to open the Assignment on Vector Algebra Chapter 10 Class 12

Topics to be discussed here

1) Scalar and Vector Quantities

2) Different types of vectors

3) Unit Vector and Method of finding unit vector

4) Addition of vectors

5) Properties of vector addition

6) Components of a Vector

7) Method of finding Position Vector :

8) Direction angles, Direction Cosines and Direction Ratios

9) Section Formula and Mid Point Formula

10) Scalar or Dot product of two vectors

11) Projection of a vector on a line

12) Vector Product (or cross product ) of two Vectors

13) Scalar Triple Product of Vectors

14) Properties of Scalar Triple Product of vectors

15) Volume of a parallelopiped using scalar Triple Product

Complete Explanation of the above mentioned topics

1) Scalar and Vector Quantities

All physical quantities are divided into two categories :- scalar quantities and vector quantities.

Scalar quantities:- Those physical quantities which possess only magnitude and no direction are called scalar quantities. For example Mass, Length, Time, Temperature , speed, Volume, Density, Work, Energy, Electric charge etc all are scalar quantities.

Vector Quantities:- Those physical quantities which possess both magnitude and direction are called vector quantities. For example : Displacement, Velocity, Momentum, Force, Weight, Acceleration etc. all are vector quantities.

Note: Quantities having magnitude and direction but do not obey the parallelogram laws of vector addition will not be treated as vectors.

For example rotation of a rigid body through a certain finite angle have both magnitude and direction but do not obey the parallelogram laws of vector addition. So it is not treated as the vector quantity.

2) Different types of vectors

Zero (or null) vector : A vector whose initial and final points are coincident is called zero or null vector. It has zero magnitude and no specific direction. Its direction is arbitrary. \[Zero\; vector\; is\; denoted\; by\;\; \overrightarrow{O}\]

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

Collinear Vectors (or Parallel Vectors) : Two vectors are said to be collinear or parallel if they have same supports or parallel supports.

Co- initial Vectors : Vectors having same initial point are called co-initial vectors.

Coplanar vectors : Three or more vectors which lie in the same plane or are parallel to the same plane are called coplanar vectors. Two vectors are always coplanar.

Co-terminous vectors : Vectors having same terminal point are called co-terminous vectors.

Like vectors : Collinear vectors having same direction are called like vectors.

Negative of a vector : Two vectors of equal magnitude are said to be negative of each other if they are of opposite direction.

Unit Vector : A vector whose magnitude is unity is called a unit vector. It is denoted by â . Magnitude of unit vector is always 1.

3) Unit Vector and Method of finding unit vector

Unit Vector : A vector whose magnitude is unity is called a unit vector. It is denoted by â . Magnitude of unit vector is always 1.

Unit vector of vector a is given by $\hat{a}=\frac{\overrightarrow{a}}{|\overrightarrow{a}|}$

Method of finding unit vector
Let any vector a is given by
$\vec{a}=x\widehat{i}+y\widehat{j}+z\widehat{k}$

Magnitude of this vector is given by
$|a|=\sqrt{x^{2}+y^{2}+z^{2}}$

Now divide the given vector by its magnitude then we get a unit vector in the direction of the given vector.

$\widehat{a}=\frac{\vec{a}}{|\vec{a}|}$

$=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\widehat{i}+\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}\widehat{j}+\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\widehat{k}$

4) Addition of vectors

Triangle law of vector addition

If two vectors are represented in magnitude and direction by two sides of a triangle taken in the same order, then their sum is represented by the third side taken in opposite order.

$\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}$

$\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

Parallelogram Law of Vector Addition : If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram which is coinitial with the given vectors.

$\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}$ or $\\\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

$\overrightarrow{OC}+\overrightarrow{CB}=\overrightarrow{OB}$ or $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

5) Properties of vector addition

Vector addition is commutative

$\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

Vector addition is associative

$\overrightarrow{a}+\left (\overrightarrow{b}+\overrightarrow{c} \right )=\left (\overrightarrow{a}+\overrightarrow{b} \right )+\overrightarrow{c}$

Existence of additive identity

$\overrightarrow{a}+\overrightarrow{O}=\overrightarrow{a} =\overrightarrow{O} + \overrightarrow{a}$

This shows that O is the additive identity for vector addition

Existence of additive inverse

$\overrightarrow{a}+(-\overrightarrow{a})=\overrightarrow{O}$

Multiplication of a vector by a scalar

$m(\overrightarrow{a}+\overrightarrow{b})=m\overrightarrow{a}+m\overrightarrow{b}$

$(m+n)\overrightarrow{a}=m\overrightarrow{a}+n\overrightarrow{a}$

6) Components of a Vector

Let us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and z-axis respectively. Then

$\left | \overrightarrow{OA} \right |=\left | \overrightarrow{OB} \right |=\left | \overrightarrow{OC} \right |=1$

The vectors $\overrightarrow{OA},\; \overrightarrow{OB},\; \overrightarrow{OC}$ each having magnitude 1 are called unit vectors and are denoted by $\hat{i},\; \hat{j},\; \hat{k}$

If |OA| = x, |OB| = y, |OC| = z then the position vector of any vector in three dimensional plane is given by

$\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$

'î' is the unit vector along x -axis,' ĵ' is the unit vector along y - axis and 'k' is the unit vector along z - axis.

If x = a, y = b, z = c then position vector becomes

$\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$

Here x component of vector r is a, y component is b and z component is c.

a, b, c are also be called the direction ratios of the vector.

7) Method of finding Position Vector :

Let P be any point in the space (or plane) having co-ordinates (x, y, z) with respect to fixed point (0, 0, 0) as origin, then the vector OP is called the position vector of point P with respect to point O it is usually denoted by vector r.

$\overrightarrow{OP}= \overrightarrow{r} = x\hat{i}+y\hat{j}+z\hat{k}$

$\left |\overrightarrow{OP} \right |=\left |\overrightarrow{r} \right |= \sqrt{x^{2}+y^{2}+z^{2}}$

Position Vector from vector A to vector B

$\overrightarrow{AB}$ Position vector of B - Position vector of A

Position vector AB passing through the points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is given by

$\overrightarrow{AB}= \left (x_{2}\hat{i}+y_{2}\hat{j}+z_{2}\hat{k} \right )-\left ( x_{1}\hat{i}+ y_{1}\hat{i}+ z_{2}\hat{k} \right )$

$\overrightarrow{AB}= \left ( x_{2}-x_{1} \right )\hat{i}+\left ( y_{2}-y_{1} \right )\hat{j}+\left ( z_{2}-z_{1} \right )\hat{k}$

8) Direction angles, Direction Cosines and Direction Ratios

Direction angles:

These are the angles made by the vector with the positive direction of the axis. These are denoted by α, β, 𝜸

Direction cosines:

Cosines of the direction angles are called direction cosines.

If α, β, 𝜸 are the direction angles made by a vector with the axis then cosα, cosβ, cos𝜸 are called the direction cosines. These are also denoted by l, m, n

l = cosα, m = cosβ, n = cos𝜸

If l, m, n are the direction cosines of a line then l²+ m²+ n² = 1

Direction ratios:

The terms which are proportional to the direction cosines are called direction ratios. These are denoted by (a, b, c)

For Example:

Let any vector $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$

Its magnitude is given by $|\overrightarrow{r}|=\sqrt{a^{2}+b^{2}+c^{2}}$

Where a, b, c are the direction ratios of the vector

The direction cosines of the given vector is given by

$\left ( \frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}},\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}},\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}} \right )$

9) Section Formula and Mid Point Formula

Section Formula (For internal division)

Let A and B are two points with position vectors, vector a and vector b and let the point P with position vector 'r' divide the line segment AB internally in m : n, then

$\overrightarrow{r}= \frac{m\overrightarrow{b} +n\overrightarrow{a}}{m+n}$

Mid Point Formula

If point P is the mid point of the line segment AB then m = n = 1 and

$\overrightarrow{r}=\frac{\overrightarrow{a}+\overrightarrow{b}}{2}$

Section Formula (For External division)

Let A and B are two points with position vectors, vector a and vector b and let the point P with position vector 'r' divide the line segment AB Externally in m : n, then

$\overrightarrow{r}= \frac{m\overrightarrow{b} -n\overrightarrow{a}}{m-n}$

Centroid of the triangle

If A, B, C are the vertices of the triangle with position vectors, vector a, vector b, vector c, then the position vector of the centroid of the triangle is given by

$\frac{\overrightarrow{a}+ \overrightarrow{b} +\overrightarrow{c}}{3}$

10) Scalar or Dot product of two vectors

Scalar or dot product of two vectors, vector a and vector b is given by

$\overrightarrow{a}. \overrightarrow{b}= |\overrightarrow{a}||\overrightarrow{b}|cos\theta$

where 𝞡 is the angle between two vectors, vector a and vector b.

$\hat{i}.\hat{i}=\hat{j}.\hat{j}=\hat{k}.\hat{k}=1$

$\hat{i}.\hat{j}=\hat{j}.\hat{k}=\hat{k}.\hat{i}=0$

Two vectors are perpendicular if

$\overrightarrow{a}.\overrightarrow{b}=0$

Two vectors are parallel if

$\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|$

Properties of scalar product

The scalar product is commutative i.e.

$\overrightarrow{a}.\overrightarrow{b}= \overrightarrow{b}.\overrightarrow{a}$

Scalar product is distributive over vector addition, i.e.

$\overrightarrow{a}.\left (\overrightarrow{b}+\overrightarrow{c} \right )=\overrightarrow{a}.\overrightarrow{b} +\overrightarrow{a}.\overrightarrow{c}$

If $\overrightarrow{a},\overrightarrow{b}$ are two vectors then: $\left | \overrightarrow{a}+\overrightarrow{b} \right |\leq \left | \overrightarrow{a} \right |\left | \overrightarrow{b} \right |$

Angle Between two Vectors

In Vector form

If θ is the angle between $\overrightarrow{a}\; \; and\; \; \overrightarrow{b}$ then

$\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|cos\theta$

$cos\theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}$

In Cartesian form

If $\overrightarrow{a}=a_{1}\hat{i}+b_{1}\hat{j}+c_{1}\hat{k}$ and

$\overrightarrow{b}=a_{2}\hat{i}+b_{2}\hat{j}+c_{2}\hat{k}$ then

$cos\theta =\left | \frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{\left ( a_{1}^{2} +b_{1}^{2}+c_{1}^{2}\right )\left ( a_{2}^{2}+b_{2}^{2}+c_{2}^{2} \right )}} \right |$

11) Projection of a vector on a line

$Projection \;\; of\; \; \overrightarrow{a} \; on\; \overrightarrow{b} = \frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{b}|}$

$Projection\; \; of\; \; \overrightarrow{b}\; on\; \overrightarrow{a}\; = \frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}|}$

$Let\; \overrightarrow{a} \; is\; denoted \; by\; \overrightarrow{AB}\; and$

$if\; \; \theta =0, \; then\; Projection\; of\; \; \overrightarrow{AB}=\overrightarrow{AB}$

$if \;\; \theta = \pi,\; then\; Projection\; of\; \overrightarrow{AB}=\overrightarrow{BA}$

$if\; \; \theta =\frac{\pi }{2},\; \; or\; \; \frac{3\pi }{2},\; \;$ then projection of $\overrightarrow{AB}$ will be zero

12) Vector Product (or cross product ) of two Vectors

Vector product of two vectors $\vec{a}\; and\; \vec{b}$ is given by   $\vec{a}\times \vec{b}=|\vec{a}||\vec{b}|sin\theta \hat{n}$

Where $\hat{n}$ is a unit vector perpendicular to both   $\vec{a}\; and\; \vec{b}$

Vector product between two vectors $\vec{a}\; and\; \vec{b}$ is given by   $\vec{a}\times \vec{b}=|\vec{a}||\vec{b}|sin\theta \hat{n}$

Where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}\; and\; \vec{b}$

$\left | \vec{a}\times \vec{b} \right |=\left | \vec{a} \right | \; \left | \vec{b} \right |sin\theta$

If two vectors $\vec{a}\; and\; \vec{b}$ are perpendicular then $\vec{a}\times \vec{b}=|\vec{a}||\vec{b}|$

Because for perpendicular vectors θ = 90^o and sin90^o = 1

If two vectors $\vec{a}\; and\; \vec{b}$ are parallel then $\vec{a}\times \vec{b}=0,$

Because for parallel vectors θ = 0 and sin 0^o = 0

$\hat{i}\times \hat{i}=\hat{j}\times \hat{j}= \hat{k} \times \hat{k}=0$

$\hat{i}\times \hat{j}=\hat{k}\: \: or\: \: \hat{j}\times \hat{i}=-\hat{k}$

$\hat{j}\times \hat{k}=\hat{i}\: \: or\: \: \hat{k}\times \hat{j}=-\hat{i}$

$\hat{k}\times \hat{i}=\hat{j}\: \: or\: \: \hat{i} \times \hat{k}=-\hat{j}$

If $\vec{a}=a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k}\; \;and\;\; \vec{b}= b_{1}\hat{i}+b_{2}\hat{j}+b_{3}\hat{k}$ then

$\vec{a}\times \vec{b}=\left | \begin{matrix} \hat{i} & \hat{j} & \hat{k}\\ a_{1} & a_{2} & a_{3}\\ b_{1}&b_{2} &b_{3} \end{matrix} \right |$

Vector product of two vectors is not commutative or

$\vec{a}\times \vec{b}=-(\vec{b}\times \vec{a})$

Area of Parallelogram

If $\vec{a}\; and\; \vec{b}$ are two vectors representing the adjacent sides of a parallelogram then

Area of parallelogram = $=\; |\overrightarrow{a}\times \overrightarrow{b}|$

If $\vec{a}\; and\; \vec{b}$ representing two diagonals of a parallelogram then

Area of parallelogram = $\frac{1}{2}\; |\overrightarrow{a}\times \overrightarrow{b}|$

Area of triangle

If $\vec{a}\; and\; \vec{b}$ are two vectors representing two sides of triangle then

Area of triangle = $\frac{1}{2} \left |\overrightarrow{a}\times \overrightarrow{b} \right |$

Lagrange's Identity : If $\vec{a}\; and\; \vec{b}$ are two vectors then

$\left | \vec{a}\times \vec{b} \right |^{2}=|\vec{a}|^{2}|\vec{b}|^{2}-(\vec{a}.\vec{b})^2$

13) Scalar Triple Product of Vectors

Definition If $\overrightarrow{a},\overrightarrow{b}, \; and \; \overrightarrow{c}$ are three vectors then $\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})$ is called the scalar triple product and is denoted by $[\overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c}]$

$[\overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c}]=\vec{a}.\left ( \vec{b}\times \vec{c} \right )\;\; or\; \; \left ( \vec{a}\times \vec{b} \right ).\vec{c}$

14) Properties of Scalar Triple Product of vectors

Property 1:In scalar triple product all the three vectors are cyclically permuted

$\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})= \overrightarrow{b}.(\overrightarrow{c}\times \overrightarrow{a})=\overrightarrow{c}.(\overrightarrow{a}\times \overrightarrow{b})$ or

$\left [\overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c} \right ]=\left [\overrightarrow{b}\; \; \overrightarrow{c}\; \; \overrightarrow{a} \right ]=\left [\overrightarrow{c}\; \; \overrightarrow{a}\; \; \overrightarrow{b} \right ]$

Property 2. Change in the order of the vectors in the scalar triple product changes the sign of the scalar triple product

$\left [\overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c} \right ]=-\left [\overrightarrow{b}\; \; \overrightarrow{a}\; \; \overrightarrow{c} \right ]=-\left [\overrightarrow{c}\; \; \overrightarrow{b}\; \; \overrightarrow{a} \right ]=-\left [\overrightarrow{a}\; \; \overrightarrow{c}\; \; \overrightarrow{b} \right ]$

Property 3. In scalar triple product the position of dot and cross can be interchanged provided that the cyclic order of the vectors remain unchanged.

$\overrightarrow{a}.\left ( \overrightarrow{b}\times \overrightarrow{c} \right )=\left (\overrightarrow{a}\times \overrightarrow{b} \right ). \overrightarrow{c}$

Property 4. Scalar triple product of three vectors is zero if any two of them are equal.

$\overrightarrow{a}.\left ( \overrightarrow{a}\times \overrightarrow{c} \right )=\overrightarrow{b}.\left ( \overrightarrow{b}\times \overrightarrow{c} \right )=\overrightarrow{a}.\left ( \overrightarrow{c}\times \overrightarrow{c} \right )=0$

Property 5. For any three vectors $\overrightarrow{a},\overrightarrow{b}, \overrightarrow{c}$ and scalar λ, we have

$\left [ \lambda \overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c} \right ]=\lambda \left [ \overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c} \right ]$

Property 6 For any three vectors $\overrightarrow{a},\overrightarrow{b}, \overrightarrow{c}$ and scalar l, m, n, then we have

$\left [ l \overrightarrow{a}\; \;m \overrightarrow{b}\; \;n \overrightarrow{c} \right ]=lmn \left [ \overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c} \right ]$

Property 7. For any four vectors $\overrightarrow{a},\overrightarrow{b}, \overrightarrow{c}\; and \; \overrightarrow{d}$ and we have

$\left [ \overrightarrow{a}+\overrightarrow{b}\; \; \overrightarrow{c}\; \; \overrightarrow{d} \right ]= \left [ \overrightarrow{a}\; \; \overrightarrow{c}\; \; \overrightarrow{d} \right ]+ \left [ \overrightarrow{b}\; \; \overrightarrow{c}\; \; \overrightarrow{d} \right ]$

Property 8. If $\overrightarrow{a},\overrightarrow{b}, \overrightarrow{c}$ are coplanar then

$\left [ \overrightarrow{a}\; \; \overrightarrow{b}\; \; \overrightarrow{c} \right ]= 0$

15) Volume of a parallelopiped using scalar Triple Product

If $\vec{a},\; \vec{b},\; \vec{c}$ are co-terminous edges of the parallelopiped then

Volume of Parallelopiped = $[\vec{a}\; \; \vec{b}\; \; \vec{c}]$

Method of calculating the Scalar Triple Product.

If $\vec{a}=a_{1}\hat{i} +a_{2}\hat{j}+ a_{3}\hat{k}$

$\vec{b}= b_{1}\hat{i}+b_{2}\hat{j}+b_{3}\hat{k}$ and

$\vec{c}= c_{1}\hat{i} +c_{2}\hat{j}+c_{3}\hat{k}\;$ then

Scalar triple product of $\vec{a},\; \vec{b},\; \vec{c}$ is given by

$\left [ \vec{a}\;\; \;\vec{b}\; \; \; \vec{c} \right ]=\left | \begin{matrix} a_{1} & a_{2} &a_{3} \\ b_{1} & b_{2} &b_{3} \\ c_{1}&c_{2} & c_{3} \end{matrix} \right |$

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