Vector Details with Diagram (Part 3)

Vector Details with Diagram (Part 3)

Vector Details with Diagram (Part 3)

Ifclip_image002[4] Thenclip_image004[4]=? clip_image006[4]clip_image008[4] clip_image010[4] clip_image012[4] clip_image014[4] clip_image016[4]

 

clip_image018[4]

 

 

 

 

Vector Details with Diagram (Part 3)

Dot product or Scalar product:

Definition:The scalar or dot product of two vectors is a scalar equal to the product of the magnitude of the vectors and the cosine of the angle between them.

Explanation: Let OA and OB represent two vectors clip_image020[16]andclip_image022[16]respectively both in magnitude and direction [Fig: Right side]. Both these vectors are inclined clip_image024[4]to each other by an angle q. Now, according to the definition of scalar product, we get, clip_image026[4] clip_image028[4]

From the figure, clip_image030[4] clip_image032[4]BM is normal on OA and AN is normal on OB

∴ OM = OB cosθ or, OM = Q cosθ ∴clip_image034[4]

= clip_image036[4](OM) = (magnitude ofclip_image020[17]) (projection of clip_image022[17]onclip_image020[18]) Thus, clip_image040[4]= (magnitude ofclip_image022[18]) (projection of clip_image020[19]onclip_image022[19]) This means that the scalar product of two vectors is the product of the magnitude of either vector and the projection of the in its direction.

Important condition about scalar product: (i) If θ = 0°, thenclip_image043[4]. This is the condition for two vectors to be parallel to each other. (ii) If θ = 90°, thenclip_image045[4]. This is the condition for two vectors to be perpendicular to each other. (iii) If θ = 180°, thenclip_image047[4]. This is the condition for two vectors to be parallel but opposite directed. clip_image049[4]

 

Dot product of rectangular unit vector: Thethree rectangular unit vectors are clip_image051[6] andclip_image053[6]are perpendicular to each other

(a)clip_image055[4]

(b)clip_image057[4]

Vector Details with Diagram

Vector Details with Diagram (Part 3)

 

Vector product or cross product:

Definition: The vector or cross product of any two vectors is another vector, the magnitude of which is obtained by multiplying the magnitudes of the constituent vectors with the sine of the angle between them and direction is perpendicular to the plane containing the vectors.

Explanation: Let clip_image020[20]and clip_image022[20]be the two vectors acting at O making an angle θ. According to definition the vector product is, clip_image061[4] Here clip_image063[8] is a unite vector which represent the direction of clip_image065[12]and. The direction of clip_image063[9]is

found from clip_image067[6]

the right-handed screw rule. clip_image063[10]is a unit vector which denoted the direction of the resultant of clip_image069[4]

clip_image071[4]If clip_image073[4]then the direction of R is perpendicular to the plane of clip_image020[21]and clip_image022[21]

clip_image077[4] clip_image079[4]

(Magnitude of P) (Magnitude of Q along the perpendicular direction of P) Again, clip_image081[4]

clip_image083[4]

clip_image085[4]

(Magnitude of Q) (Magnitude of P along the perpendicular direction of Q)

clip_image087[4]I.e. Cross product does dos obey commutative law.

Important condition about vector product:

(i) When θ = 0°, thenclip_image089[4]. This is the condition that two vectors are parallel to each other.

(ii) When θ = 90°, thenclip_image091[4]. This is the condition that two vectors are perpendicular to each other.

(iii) When θ = 180°, thenclip_image093[4]. This is the condition that two vectors are parallel and opposite to each other. clip_image094[4]

Cross product or vector product of rectangular unit vector:

Thethree rectangular unit vectors are clip_image051[7] andclip_image053[7]are perpendicular to each other

(a) clip_image096[4]

(b) clip_image098[4]

 

The scalar or dot product obeys commutative law but Cross product does dos obey commutative law:

Let the angle between the vectors clip_image020[22]andclip_image022[22]is a, then clip_image102[4] Again, clip_image104[4] From (1) and (2) we get, clip_image106[4] i.e. the scalar or dot product obeys

Vector Details with Diagram (Part 3)

commutative law.

On the other hand the direction of clip_image108[4]andclip_image110[4]are opposite of each other, but the magnitude is the same, i.e.clip_image112[4] And clip_image114[4] clip_image116[4] From (3) and (4) we get, clip_image118[4] clip_image120[4] I.e. the cross product does dos obey commutative law.

Commutative Law:

If clip_image122[4]are two vector of same nature then

clip_image124[4]clip_image126[4]This is Commutative Law.

Proof:Suppose ORQP is a parallelogram with diagonal OQ and clip_image128[4]; clip_image130[4]

From ∆OPQ, clip_image132[4] And from ∆ORQ,clip_image134[4]

From (1) and (2) we get clip_image136[4] I.e.clip_image138[4]

 

Associative Law:

If clip_image140[4]are three vectors of same physical nature, then, clip_image142[4]clip_image144[4]this is the Associative law for vector addition. Proof: Suppose clip_image146[4],clip_image148[4]andclip_image150[4].

Appling triangle rule we can write from ∆ OPQ, clip_image152[4] and from ∆ PQR, clip_image154[4] and from ∆ OQR, clip_image156[4]I.e. clip_image158[4] again from ∆ OQR, clip_image160[4] I.e. clip_image162[4] clip_image164[4]This is Associative law:

 

Distributive law: Distributive law of scalar product is clip_image166[4]clip_image168[4]

Proof: Suppose clip_image170[4]are three vectors clip_image172[4] Now From the figure, we get, clip_image174[4] clip_image176[4] clip_image178[4] clip_image180[4] clip_image182[4] clip_image184[4] clip_image186[4] clip_image188[4] clip_image190[4]

The equation of a position vector in three dimensional reference system i.e. proof of clip_image192[4]:

Suppose, OX, OY and OZ be three lines perpendicular to each other and represent X, Y and Z axis respectively. clip_image194[4]Is a position vector which is represented by line OP, i.e. clip_image196[4]. The co-ordinate of P is (x,y,z) and clip_image198[4],clip_image200[4]and clip_image202[4]are the unit vectors along the axis X, Y and Z respectively. PN is drawn normal to plane XY and PQ is normal to OZ. ON is joined. NL and NM is drawn normal to OX and OY respectively. From figure, OL = x, OM = y, NP = z; By the triangle Law, clip_image204[4] clip_image206[4] clip_image208[4] clip_image210[4] clip_image212[4]

Again From figure we get, OP2=ON2+NP2

clip_image214[6] OP2 = OL2+LN2+NP2

clip_image214[7] OP2 = OL2+OM2+OQ2

clip_image216[4] clip_image218[4]

 

Vector Resolution: A vector quantity can be resolved into two or more vectors in different directions. Each resolving vector is known as component of the original vector. The process of resolving a vector into two or more vector is called vector resolution or resolution of vector. Let clip_image065[13] be a vector acting at a point O, along OC. The vector clip_image065[14]is to be resolved into two components. Let clip_image221[4]and clip_image223[4]be the two resolved components acting along OA and OB respectively making at angle α and β with the resultant vectorclip_image065[15]. Let us complete the parallelogram OACB. Now from the law of triangle of addition, we get, clip_image225[4]i.e. clip_image227[4] clip_image229[4]To express the com

ponent vector in scalar form, we get,

From the sine law of triangle, clip_image231[4] clip_image233[4] clip_image235[4] clip_image237[4] andclip_image239[4]

Now, if the vectorresolved in to perpendicularly, then α + β = 90°

∴ sin (α + β) = sin 90° = 1 and α + β = 90° ∴ β = 90°- α

and

∴ sin β = sin (90 – α) = cos α therefore, P = R sin β

∴P = R cos α and Q = R sin α

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Vector Details with Diagram (Part 3)

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Vector Details with Diagram (Part 1)

Vector Details with Diagram (Part 3)

Vector Details with Diagram (Part 3)

Vector Details with Diagram (Part 3)

Shah JamalEnglish VersionHSCNotePhysics 1st Paperassociative law,commutative law,cross product,distributive law,product,scalar product,the equation of a position vector,vector product,vector ResolutionVector Details with Diagram (Part 3) Vector Details with Diagram (Part 3) If Then=?           Vector Details with Diagram (Part 3) Dot product or Scalar product: Definition:The scalar or dot product of two vectors is a scalar equal to the product of the magnitude of the vectors and the...Shah Jamal's Online Classroom