Laws of Vector

Multiplying a vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>&#x2192;</mo></mover></math> with a positive number n gives a vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>&#x2192;</mo></mover></math> (= n<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>&#x2192;</mo></mover></math>) whose magnitude is changed by the factor n, but the direction is same as that <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>&#x2192;</mo></mover></math>.

Also, Multiplying a vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>&#x2192;</mo></mover></math> with a negative number n gives a vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>&#x2192;</mo></mover></math> (= – n<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>&#x2192;</mo></mover></math>) whose direction is opposite to the direction of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>&#x2192;</mo></mover></math> and its magnitude is – n times <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mover><mi>p</mi><mo>&#x2192;</mo></mover></mfenced></math>.

An example is given:

Diagram (a): Multiplication by a positive number

Diagram (b): Multiplication by a negative number

Figure:6.a

Vector addition and subtraction – graphical method:

Let OA = p be a vector. From the terminal point of p, another vector AB = q is drawn. Then, the vector OB from the initial point of p to the terminal point B of b is called the sum of vectors p and q and is denoted by s = p + q. This is called the triangle law of addition of vectors.

Figure:6.b

From the above diagram, we get,

s = p + q

Or,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>O</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>=</mo><mover><mrow><mi>O</mi><mi>A</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover></math>

Points to remember:

  1. Vector addition is commutative i.e. p + q = q + p
  2. Vector addition is associative i.e. p + q + s = (p + q) + s = p + (q + s) = (p + s) + q
  3. p + 0 = p
  4. p + (- p) = 0

Now, the above diagram can be expressed as below:

Figure:6.c

The direction of vector b is reversed and then the vector addition process is followed.

Then, it can be written as,

s = p – q = p + (- q)

Vector addition – analytical method:

Parallelogram law: If two vectors P and Q are represented by two adjacent sides of a parallelogram both pointing outwards, then the diagonal of the parallelogram represents the resultant of P and Q.

Figure:6.d

From the diagram, we get,

R = P + Q

BC is normal to OC.

From the geometry of the diagram,

$OB^{2}=OC^{2}+CB^{2}$ ……………………….. (1)

Now, AC = $Qcos\theta $

So, OC = P + $Qcos\theta $

And BC = $Qsin\theta $

From equation (1),

$R^{2}=(P + Qcos\theta )^{2}+$$(Qsin\theta )^{2}$

Or, $R=\sqrt{P^{2}+Q^{2}+2PQcos\theta }$……………………….. (2)

The direction of the resultant vector R from vector A is given by,

$\tan \varphi =BC/OC$

Or, $\tan \varphi =\frac{Qsin\theta }{P + Qcos\theta }$

Or, $\varphi =\tan ^{-1}(\frac{Qsin\theta }{P + Qcos\theta })$ ……………….. (3)

Equation (2) and (3) give the magnitude and direction of the resultant vector respectively.

Points to remember:

  1. As $R=\sqrt{P^{2}+Q^{2}+2PQcos\theta }$, R will be maximum when $\cos \theta $= 1 i.e. $\theta =0$. This signifies that vectors are parallel. So, $R_{\max }=P+Q$.
  2. R will be minimum when $\cos \theta $= – 1 i.e. $\theta =180$. This signifies that vectors are antiparallel. So, $R_{\min }=P-Q$.

Zero vector:

Zero vector is a vector whose initial and terminal points are same. The magnitude of the zero vector is always zero. This is also called null vector. The zero vector is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mn>0</mn><mo>&#x2192;</mo></mover></math>.

When a zero vector is added or subtracted from a vector, then the result is the vector itself in each case.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>R</mi><mo>&#x2192;</mo></mover><mo>+</mo><mover><mn>0</mn><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>R</mi><mo>&#x2192;</mo></mover><mspace linebreak="newline"/><mover><mi>R</mi><mo>&#x2192;</mo></mover><mo>-</mo><mover><mn>0</mn><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>R</mi><mo>&#x2192;</mo></mover></math>

Resolution of vectors:

If any two nonzero vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> are in a plane with different directions and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>A</mi><mo>&#x2192;</mo></mover></math> be another vector in the same plane, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>A</mi><mo>&#x2192;</mo></mover></math> can be expressed as a sum of two vectors – one multiplying <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover></math> by a real number and the other by another real number.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>A</mi><mo>&#x2192;</mo></mover><mo>=</mo><mi>&#x3B1;</mi><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>+</mo><mi>&#x3B2;</mi><mover><mi>b</mi><mo>&#x2192;</mo></mover></math>, where $\alpha $and $\beta $ and are real numbers

So, vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>A</mi><mo>&#x2192;</mo></mover></math> has been resolved into two component vectors i.e. and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3B2;</mi><mover><mi>b</mi><mo>&#x2192;</mo></mover></math>.

Resolution in two dimensions:

Consider a vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover></math> vector lies in XY plane such that, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>=</mo><mover><msub><mi>a</mi><mi>x</mi></msub><mo>&#x2192;</mo></mover><mo>+</mo><mover><msub><mi>a</mi><mi>y</mi></msub><mo>&#x2192;</mo></mover></math>

Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>a</mi><mi>x</mi></msub><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>a</mi><mi>x</mi></msub><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mover><msub><mi>a</mi><mi>y</mi></msub><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>a</mi><mi>y</mi></msub></math>
, where are unit vectors along the X and Y – axis

Figure:6.e

So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x21C0;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><msub><mi>a</mi><mi>x</mi></msub><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><msub><mi>a</mi><mi>y</mi></msub></math>

Or, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>=</mo><mover><mi>i</mi><mo>^</mo></mover><mi>a</mi><mi>cos</mi><mi>&#x3B8;</mi><mo>+</mo><mover><mi>j</mi><mo>^</mo></mover><mo>&#xA0;</mo><mi>b</mi><mi>sin</mi><mi>&#x3B8;</mi></math>

The magnitude $\vert a\vert =\sqrt{a_{x}^{2}+a_{y}^{2}}$

And direction, $\theta =tan^{-1}(a_{y}/a_{x})$

Multiplication of vectors:

Dot product:

The scalar product or dot product of any two vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> is written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>.</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math>.

The dot product is equal to the product of their magnitudes with cosine of angle between them.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>.</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover><mo>=</mo><mi>a</mi><mi>b</mi><mo>&#xA0;</mo><mi>cos</mi><mi>&#x3B8;</mi></math>

Scalar product will be maximum when $\cos \theta $= 1, so, vectors are parallel.

Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mrow><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>.</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></mrow></mfenced><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>a</mi><mi>b</mi></math>

Component of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> along c = $b cos\theta $

Or,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mover><mrow><mi>a</mi><mo>.</mo></mrow><mo>&#x2192;</mo></mover><mover><mi>b</mi><mo>&#x2192;</mo></mover></mrow><mi>a</mi></mfrac><mo>=</mo><mover><mi>a</mi><mo>^</mo></mover><mo>.</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math>

Similarly, the component of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover></math> along <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mover><mrow><mi>a</mi><mo>.</mo></mrow><mo>&#x2192;</mo></mover><mover><mi>b</mi><mo>&#x2192;</mo></mover></mrow><mi>b</mi></mfrac><mo>=</mo><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>.</mo><mover><mi>b</mi><mo>^</mo></mover></math>

 

Cross product:

The vector product or cross product of any two vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> is denoted by  <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>&#xD7;</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> .

It is defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>&#xD7;</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover><mo>=</mo><mi>a</mi><mi>b</mi><mo>&#xA0;</mo><mi>sin</mi><mi>&#x3B8;</mi><mo>&#xA0;</mo><mover><mi>n</mi><mo>^</mo></mover></math>

Where $\theta $ is the angle between <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> and ${n}$ is the unit whose direction is given by the right-hand thumb rule.

Figure:6.f

Stretch your right palm and pace perpendicular to the plane of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> in such a way that the finger is along the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>&#x2192;</mo></mover></math> and towards  <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>b</mi><mo>&#x2192;</mo></mover></math> when fingers are closed. The direction of thumb gives the direction of ${n}$.

The vector product of two vectors is always a vector perpendicular to the plane containing the two vectors.

The vector product will be maximum if $\sin \theta $=1 or $\theta =90^{0}.$

 

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