Revision: Vectors Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

 The square of a vector a, i.e., \[\vec{a^2}\] is a scalar which denotes the square of the length of a and is equal to the square of its modulus.

\[\vec{a^2}\] = \[|\vec{a}|^2\] 

A directed line segment is a line segment with an arrowhead showing direction. Its two endpoints are distinguishable as the initial point and the terminal point

The vector is denoted by \[\overrightarrow{AB}\]  

The vector drawn from the origin O(0,0,0)to a point P(x,y,z) is called the position vector of the point P.

It is denoted by: \[\vec{OP}=x\hat{i}+y\hat{j}+z\hat{k}\]

Magnitude of Position Vector: \[|\vec{OP}|=\sqrt{x^2+y^2+z^2}\]

When quantities can be represented by a certain number of units with no association with direction in space, they are called scalar quantities and numbers that represent them are called scalars. 

The scalar product or inner product of two non-zero vectors written as like \[\mid a\mid\mid b\mid\cos\theta\]\[\vec{a}\], \[\vec{b}\] is defined to be the scalar \[\left|\vec{a}\right|\left|\vec{b}\right|\cos\theta\] = \[ab\cos\theta\]

where a \[=|\vec{a}|\], b = \[=|\vec{b}|\] and θ = (0 ≤ θ ≤ π) is the angle between\[\vec{a}\] and \[\vec{b}\].

Let \[\hat{i}\],\[\hat{j}\], \[\hat{k}\] be unit vectors in the positive direction of the three mutually perpendicular coordinate axes, x-axis,  y-axis and z-axis, respectively. Then, these vectors are said to form an orthonormal triad of vectors. 

Dot Products: 

• \[\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1\]

• \[\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0\]

If a vector \[\overrightarrow{AB}\] is denoted by \[\overrightarrow{a}\], then \[\mid\overrightarrow{a}\mid\] denotes the positive length of the vector a, also called the magnitude or norm or modulus of the vector.

Thus \[\left|\vec{a}\right|\] = a, if a is the positive length of \[\overrightarrow{a}\].

\[\mid\overset{\rightarrow}{\operatorname*{\mathbf{AB}}}\mid=\mid\overset{\rightarrow}{\operatorname*{a}}\mid=a\]

A quantity which has both magnitude and direction is called a vector quantity, provided that two such quantities can be combined by vector addition.

A vector quantity can be represented in the plane by a directed line segment, whose length is proportional to the magnitude of the vector and whose direction is the direction of the vector.

When the direction of rotation is anticlockwise, then the rotation will move the screw upwards. It is called a right-handed orientation or a right-handed screw rule. 

Two vectors \[\vec{a}\] and \[\vec{b}\] are parallel if one is a scalar multiple of the other.

\[\vec{a}=\lambda\vec{b}\] (λ is a scalar)

The cosines of the angles made by a vector with the positive directions of the coordinate axes are called the direction cosines of the vector.

If a vector \[\vec{a}\] makes angles α,β,γ with the positive x, y and z axes respectively, then:

l = cosα, m = cosβ, n = cosγ

are called the direction cosines of the vector.

In Cartesian Form:

\[l=\frac{x}{r},\quad m=\frac{y}{r},\quad n=\frac{z}{r}\]

For any three given vectors, the scalar product of one of the vectors and the cross product of the remaining two, is called a scalar triple product

Thus, \[\vec{a},\vec{b},\vec{c}\] are three vectors, then \[(\vec{a}\times\vec{b})\cdot\vec{c}\]is called the scalar triple product and is denoted by \[[\vec{a}\vec{b}\vec{c}]\mathrm{~or~}[a,b,c]\]

Any three numbers proportional to direction cosines are called direction ratios.

• Denoted by a, b, c

• A line has infinitely many direction ratios.

\[l=\frac{a}{\sqrt{a^2+b^2+c^2}}\], \[m=\frac{b}{\sqrt{a^2+b^2+c^2}}\], \[n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]

Let\[\vec{a}\] and \[\vec{b}\]be two non-zero, non-parallel vectors, and let θ be the angle between them such that (0 < θ < π).

\[\vec{a}\times\vec{b}=\left|\vec{a}\right|\left|\vec{b}\right|\sin\theta\left.\hat{n}\right.\]

or

\[\vec{a}\times\vec{b}=ab\sin\theta\mathrm{~}\hat{n}\]

where \[\hat{n}\] is a unit vector perpendicular to both \[\vec{a}\] and\[\vec{b}\] such that\[\vec{a}\], \[\vec{b}\], \[\hat{n}\] form a righthanded triad of vectors.

\[(a\hat{i}+b\hat{j})+(x\hat{i}+y\hat{j})=(a+x)\hat{i}+(b+y)\hat{j}\]

\[\lambda(a\hat{i}+b\hat{j})=\lambda a\hat{i}+\lambda b\hat{j}\]

\[\vec{OR}=\frac{m\vec{q}+n\vec{p}}{m+n}\]

\[\vec{OR}=\frac{m\vec{q}-n\vec{p}}{m-n}\]

Projection vector of \[\vec{a} on \vec{b} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right) \vec{b}, \vec{b} \neq \vec{0}\]

Projection vector of \[\vec{b} on \vec{a} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right) \vec{a}, \vec{a} \neq \vec{0}\]

cosθ \[= \frac{ \vec{a} \cdot \vec{b} }{ | \vec{a} | \, | \vec{b} | } = \frac{ \text{scalar product of the two vectors} }{ \text{product of their moduli} }\]

\[\vec{AB}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}\]

\[|\vec{AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

\[\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}\]

In 2D:

If \[\vec{a}=a_1\hat{i}+a_2\hat{j},\quad\vec{b}=b_1\hat{i}+b_2\hat{j}\]

\[\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2\]

Angle Between Two Vectors (2D):

\[\cos\theta=\frac{a_1b_1+a_2b_2}{\left|\vec{a}\right|\left|\vec{b}\right|}\]

In 3D

If  \[\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k},\quad\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\]

\[\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2+a_3b_3\]

1. Vector area of a triangle:

\[\text{Vector area of }\triangle ABC=\frac{1}{2}(\vec{AB}\times\vec{AC})\]

2. Collinearity condition

\[\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=\vec{0}\]

3. Area Parallelogram

\[\text{Area of parallelogram}=|\vec{a}\times\vec{b}|\]

If \[\mathrm{M}({\overline{m}})\] is the mid-point of the line segment joining the points \[\mathrm{A}({\overline{a}})\] and \[\mathrm{B}({\overline{b}})\] then \[\overline{m}=\frac{\left(\overline{a}+\overline{b}\right)}{2}\]

Scalar projection = \[\frac{\text{scalar product}}{\text{Modulus of vector}}\] 

\[\text{Scalar Projection of }\overline{b}\mathrm{~on~}\overline{a}=\frac{\overline{a}\cdot\overline{b}}{|\overline{a}|}\]

\[\text{Vector Projection of }\overline{b}\mathrm{~on~}\overline{a}=\left(\overline{a}\cdot\overline{b}\right)\frac{\overline{a}}{\left|\overline{a}\right|^{2}}\]

Statement: 

If\[\vec{a},\vec{b},\vec{c},\] are three non-zero vectors and \[\vec{a}\times\vec{c}=\vec{b}\times\vec{c}\], then either \[\vec{a}=\vec{b}\mathrm{~or~}(\vec{a}-\vec{b})\] and \[\vec{c}\]are parallel vectors. 

\[\vec{a}\times\vec{c}=\vec{b}\times\vec{c}\Rightarrow\vec{a}=\vec{b}\mathrm{~or~}(\vec{a}-\vec{b})\parallel\vec{c}\]

The cross product has no cancellation law

If three points O, A, and B are so chosen that  \[\overrightarrow{OA}\] and  \[\overrightarrow{AB}\] respectively represent \[\overrightarrow{a}\] and \[\overrightarrow{b}\], then \[\overrightarrow{OB}\] is defined as the sum of  \[\overrightarrow{a}\] and \[\overrightarrow{b}\] and is written as \[\overrightarrow{c}=\overrightarrow{a}+\overrightarrow{b}\], where  \[\overrightarrow{c}\] stands for the vector \[\overrightarrow{OB}\]. \[\overrightarrow{c}\] or \[\overrightarrow{a}\] + \[\overrightarrow{b}\] is also called the resultant of  \[\overrightarrow{a}\] and \[\overrightarrow{b}\]. This is known as the Triangle law of vectors.

The result of adding two co-initial vectors is the vector represented by the diagonal of the parallelogram formed with the component vectors as adjacent sides. This is the Parallelogram Law of addition of vectors, which is thus a direct consequence of the triangle law.

Statement:

Two vectors in a plane are equal iff their x-components and y-components are equal.

If \[\vec{a}=a_1\hat{i}+a_2\hat{j},\quad\vec{b}=b_1\hat{i}+b_2\hat{j}\] Then \[\vec{a}=\vec{b}\] ⟺a_{1 ​}= b₁​ and a₂​ = b₂​

If\[\vec{a}\] and \[\vec{b}\] are two non-zero vectors, then \[\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\]

Cases:

• Acute angle (0< θ < \[\frac{\pi}{2}\])
  cos⁡θ > 0  ⇒  \[\vec{a}\cdot\vec{b}>0\]
  

• Right angle (θ = \[\frac{\pi}{2}\]
  cos⁡θ = 0  ⇒  \[\vec{a}\cdot\vec{b}=0\]
  

• Obtuse angle (\[\frac{\pi}{2}\] < θ ≤ π)

  cos⁡θ < 0  ⇒  \[\vec{a}\cdot\vec{b}<0\]

• Position of dot & cross doesn’t matter
  \[(\vec{a}\times\vec{b})\cdot\vec{c}=\vec{a}\cdot(\vec{b}\times\vec{c})\]
  

• Cyclic order unchanged ⇒ STP unchanged
  \[[\vec{a}\operatorname{\vec{b}}\vec{c}]=[\vec{b}\operatorname{\vec{c}}\vec{a}]=[\vec{c}\operatorname{\vec{a}}\vec{b}]\]
  

• Interchanging two vectors changes the sign
  \[[\vec{a}\operatorname{\vec{b}}\vec{c}]=-\left[\vec{b}\operatorname{\vec{a}}\vec{c}\right]\]
  

• If any two vectors are equal
  \[[\vec{a}\operatorname{\vec{a}}\vec{b}]=0\]

• If any two vectors are parallel
  \[[\vec{a}\operatorname{\vec{b}}\operatorname{\vec{c}}]=0\]

If: \[\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\]

Then:

Addition: \[\vec{a}+\vec{b}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}\]

Scalar Multiplication: \[\lambda\vec{a}=\lambda a_1\hat{i}+\lambda a_2\hat{j}+\lambda a_3\hat{k}\]

If \[\vec{a}\] and \[\vec{b}\] are two vectors,

\[\vec{a}-\vec{b}=\vec{a}+(-\vec{b})\]

Let\[\vec{a}\] be a vector and m a scalar. Then m\[\vec{a}\] is called the product of  \[\vec{a}\] by the scalar m.

Properties:

• The direction of m\[\overrightarrow{a}\] is the same as or parallel to that of \[\overrightarrow{a}\].

• The magnitude of m\[\overrightarrow{a}\] is given by

  \[|m\vec{a}|=|m||\vec{a}|\]

• The sense of m→a is:

  • same as \[\vec{a}\], if m is positive

  • opposite to \[\vec{a}\], if m is negative