Download - 1 - Scalars & vectors
![Page 1: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/1.jpg)
SCALARS&
VECTORS
![Page 2: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/2.jpg)
What is Scalar?
![Page 3: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/3.jpg)
Length of a car is 4.5 mphysical quantity magnitude
![Page 4: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/4.jpg)
Mass of gold bar is 1 kgphysical quantity magnitude
![Page 5: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/5.jpg)
physical quantity magnitude
Time is 12.76 s
![Page 6: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/6.jpg)
Temperature is 36.8 C physical quantity magnitude
![Page 7: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/7.jpg)
A scalar is a physical quantity thathas only a magnitude.
Examples:• Mass• Length• Time
• Temperature• Volume• Density
![Page 8: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/8.jpg)
What is Vector?
![Page 9: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/9.jpg)
California NorthCarolina
Position of California from North Carolina is 3600 km in west
physical quantity magnitudedirection
![Page 10: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/10.jpg)
Displacement from USA to China is 11600 km in east
physical quantity magnitudedirection
USA China
![Page 11: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/11.jpg)
A vector is a physical quantity that has both a magnitude and a direction. Examples:• Position• Displacement• Velocity
• Acceleration• Momentum• Force
![Page 12: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/12.jpg)
Representation of a vector
Symbolically it is represented as
![Page 13: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/13.jpg)
Representation of a vector
They are also represented by a single capitalletter with an arrow above it.
P⃗B⃗A⃗
![Page 14: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/14.jpg)
Representation of a vector
Some vector quantities are represented by theirrespective symbols with an arrow above it.
F⃗v⃗r⃗
Position velocity Force
![Page 15: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/15.jpg)
Types of Vectors
(on the basis of orientation)
![Page 16: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/16.jpg)
Parallel Vectors
Two vectors are said to be parallel vectors, if they have same direction.
A⃗ P⃗
Q⃗B⃗
![Page 17: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/17.jpg)
Equal VectorsTwo parallel vectors are said to be equal vectors,
if they have same magnitude.
A⃗B⃗
P⃗
Q⃗
A⃗=B⃗ P⃗=Q⃗
![Page 18: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/18.jpg)
Anti-parallel Vectors
Two vectors are said to be anti-parallel vectors,if they are in opposite directions.
A⃗ P⃗
Q⃗B⃗
![Page 19: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/19.jpg)
Negative Vectors
Two anti-parallel vectors are said to be negativevectors, if they have same magnitude.
A⃗B⃗
P⃗
Q⃗
A⃗=− B⃗ P⃗=− Q⃗
![Page 20: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/20.jpg)
Collinear VectorsTwo vectors are said to be collinear vectors,
if they act along a same line.
A⃗
B⃗
P⃗
Q⃗
![Page 21: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/21.jpg)
Co-initial VectorsTwo or more vectors are said to be co-initialvectors, if they have common initial point.
B⃗A⃗C⃗
D⃗
![Page 22: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/22.jpg)
Co-terminus VectorsTwo or more vectors are said to be co-terminusvectors, if they have common terminal point.
B⃗A⃗
C⃗D⃗
![Page 23: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/23.jpg)
Coplanar VectorsThree or more vectors are said to be coplanar
vectors, if they lie in the same plane.
A⃗
B⃗ D⃗
C⃗
![Page 24: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/24.jpg)
Non-coplanar VectorsThree or more vectors are said to be non-coplanar
vectors, if they are distributed in space.
B⃗C⃗
A⃗
![Page 25: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/25.jpg)
Types of Vectors
(on the basis of effect)
![Page 26: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/26.jpg)
Polar Vectors
Vectors having straight line effect arecalled polar vectors.
Examples:
• Displacement• Velocity
• Acceleration• Force
![Page 27: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/27.jpg)
Axial Vectors
Vectors having rotational effect arecalled axial vectors.
Examples:• Angular momentum• Angular velocity
• Angular acceleration• Torque
![Page 28: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/28.jpg)
VectorAddition
(Geometrical Method)
![Page 29: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/29.jpg)
Triangle Law
A⃗B⃗
A⃗B⃗
C⃗
= +
![Page 30: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/30.jpg)
Parallelogram Law
A⃗ B⃗
A⃗ B⃗
A⃗ B⃗
B⃗ A⃗
C⃗
= +
![Page 31: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/31.jpg)
Polygon Law
A⃗
B⃗
D⃗
C⃗
A⃗ B⃗
C⃗
D⃗
E⃗
= + + +
![Page 32: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/32.jpg)
Commutative PropertyA⃗
B⃗B⃗
A⃗C⃗
C⃗
Therefore, addition of vectors obey commutative law.
= + = +
![Page 33: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/33.jpg)
Associative Property
B⃗A⃗
C⃗
Therefore, addition of vectors obey associative law.
= + + = + +
D⃗ B⃗
A⃗
C⃗
D⃗
![Page 34: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/34.jpg)
Subtraction of vectors
B⃗A⃗
−B⃗
A⃗
- The subtraction of from vector is defined as
the addition of vector to vector .
- = +
![Page 35: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/35.jpg)
VectorAddition
(Analytical Method)
![Page 36: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/36.jpg)
Magnitude of Resultant
P⃗O
B
A
CQ⃗ R⃗
M
θθ ⇒
OC2=OM 2+C M 2
OC2=(O A+AM )2+CM 2
OC2=OA2+2OA× AM+A C2
R2=P2+2 P×Q cos θ+Q2
R=√P2+2 PQ cosθ+Q2
![Page 37: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/37.jpg)
Direction of Resultant
P⃗O A
CQ⃗ R⃗
Mα θ
⇒
⇒
B
![Page 38: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/38.jpg)
Case I – Vectors are parallel ()P⃗ Q⃗ R⃗
R=√P2+2 PQ cos0 °+Q 2
R=√P2+2 PQ+Q2
R=√(P+Q)2
+¿ ¿
Magnitude: Direction:
R=P+Q
![Page 39: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/39.jpg)
Case II – Vectors are perpendicular ()
P⃗ Q⃗ R⃗
R=√P2+2 PQ cos90 °+Q 2
R=√P2+0+Q2
+¿ ¿
Magnitude: Direction:
α=tan−1(QP )
P⃗
Q⃗
R=√P2+Q 2
α
![Page 40: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/40.jpg)
Case III – Vectors are anti-parallel ()
P⃗ Q⃗ R⃗
R=√P2+2 PQ cos180 °+Q 2
R=√P2−2 PQ+Q2
R=√(P−Q)2
− ¿
Magnitude: Direction:
R=P−Q
If P>Q :
If P<Q :
![Page 41: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/41.jpg)
Unit vectorsA unit vector is a vector that has a magnitude of
exactly 1 and drawn in the direction of given vector.
• It lacks both dimension and unit.• Its only purpose is to specify a direction in
space.
A⃗�̂�
![Page 42: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/42.jpg)
• A given vector can be expressed as a product of its magnitude and a unit vector.
• For example may be represented as,
A⃗=A �̂�
Unit vectors
= magnitude of = unit vector along
![Page 43: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/43.jpg)
Cartesian unit vectors
𝑥
𝑦
𝑧
�̂�
�̂�
�̂�
− 𝑦
−𝑥
−𝑧
-
-
-
![Page 44: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/44.jpg)
Resolution of a VectorIt is the process of splitting a vector into two or more
vectors in such a way that their combined effect is same as that of
the given vector.
𝑡
𝑛
A⃗𝑡
A⃗𝑛A⃗
![Page 45: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/45.jpg)
Rectangular Components of 2D Vectors
A⃗
A⃗𝑥
A⃗𝑦
O 𝑥
𝑦A⃗
A⃗=A 𝑥 �̂�+A 𝑦 �̂�θ
θA𝑥 �̂�
A𝑦 �̂�
![Page 46: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/46.jpg)
A
A𝑥
A𝑦
θ
A𝑦=A sin θ
A𝑥=A cosθ
⇒
⇒
Rectangular Components of 2D Vectors
![Page 47: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/47.jpg)
Magnitude & direction from components
A⃗=A 𝑥 �̂�+A 𝑦 �̂�
A=√ A𝑥2+A𝑦
2
Magnitude:
Direction:
θ= tan−1( A 𝑦
A𝑥)θ
A𝑥
A𝑦A
![Page 48: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/48.jpg)
Rectangular Components of 3D Vectors
𝑥
𝑧
𝑦
A⃗A⃗𝑦
A⃗′A⃗𝑧
A⃗𝑥
A⃗=A⃗′+ A⃗ 𝑦
A⃗=A⃗ 𝑥+ A⃗ 𝑧+ A⃗ 𝑦
A⃗=A⃗ 𝑥+ A⃗ 𝑦+ A⃗𝑧
A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�
![Page 49: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/49.jpg)
Rectangular Components of 3D Vectors
𝑥
𝑧
𝑦
𝛼
A
A𝑥
A𝑥=A cos𝛼
![Page 50: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/50.jpg)
Rectangular Components of 3D Vectors
𝑥
𝑧
𝑦
AA𝑦𝛽
A𝑦=A cos 𝛽
![Page 51: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/51.jpg)
Rectangular Components of 3D Vectors
𝑥
𝑧
𝑦
A
A𝑧
𝛾
A𝑧=A cos𝛾
![Page 52: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/52.jpg)
Magnitude & direction from components
A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�
A=√ A𝑥2+A𝑦
2 +A𝑧2
Magnitude:
Direction:
𝛼=cos−1( A𝑥
A )𝛽=cos− 1( A 𝑦
A )𝛾=cos−1( A𝑧
A )
A
A𝑧
𝛾𝛽𝛼 A𝑥
A𝑦
![Page 53: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/53.jpg)
Adding vectors by components
A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�B⃗=B𝑥 �̂�+B𝑦 �̂�+B𝑧 �̂�
Let us have
t hen
R𝑦=(A ¿¿ 𝑦+B𝑦)¿
![Page 54: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/54.jpg)
Multiplyingvectors
![Page 55: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/55.jpg)
Multiplying a vector by a scalar
• If we multiply a vector by a scalar s, we get a new vector.
• Its magnitude is the product of the magnitude of and the absolute value of s.
• Its direction is the direction of if s is positive but the opposite direction if s is negative.
![Page 56: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/56.jpg)
Multiplying a vector by a scalarIf s is positive:
A⃗2 A⃗
If s is negative:A⃗
−3 A⃗
![Page 57: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/57.jpg)
Multiplying a vector by a vector
• There are two ways to multiply a vector by a vector:
• The first way produces a scalar quantity and called as scalar product (dot product).
• The second way produces a vector quantity and called as vector product (cross product).
![Page 58: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/58.jpg)
Scalar product
A⃗
B⃗θ
![Page 59: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/59.jpg)
Examples of scalar product
W=F⃗ ∙ s⃗
W = work done F = force s = displacement
P=F⃗ ∙ v⃗
P = power F = force v = velocity
![Page 60: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/60.jpg)
Geometrical meaning of Scalar dot product
A dot product can be regarded as the product of two quantities:
1.The magnitude of one of the vectors
2.The scalar component of the second vector along the direction of the first vector
![Page 61: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/61.jpg)
Geometrical meaning of Scalar product
θ
AB cosθ
θ
A
A cos θBB
![Page 62: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/62.jpg)
Properties of Scalar product
1The scalar product is commutative.
![Page 63: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/63.jpg)
Properties of Scalar product
2The scalar product is distributive over
addition.
A⃗ ∙ (B⃗+C⃗ )=A⃗ ∙ B⃗+A⃗ ∙ C⃗
![Page 64: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/64.jpg)
Properties of Scalar product
3The scalar product of two perpendicular
vectors is zero.
A⃗ ∙ B⃗=0
![Page 65: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/65.jpg)
Properties of Scalar product
4The scalar product of two parallel vectors
is maximum positive.
A⃗ ∙ B⃗=A B
![Page 66: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/66.jpg)
Properties of Scalar product
5The scalar product of two anti-parallel
vectors is maximum negative.
A⃗ ∙ B⃗=− A B
![Page 67: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/67.jpg)
Properties of Scalar product
6The scalar product of a vector with itselfis equal to the square of its magnitude.
A⃗ ∙ A⃗=A2
![Page 68: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/68.jpg)
Properties of Scalar product
7The scalar product of two same unit vectors is
one and two different unit vectors is zero.
�̂� ∙ �̂�= �̂� ∙ �̂�=�̂� ∙ �̂�=(1)(1)cos 0 °=1
�̂� ∙ �̂�= �̂� ∙ �̂�=�̂� ∙ �̂�=(1)(1)cos 90 °=0
![Page 69: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/69.jpg)
Calculating scalar product using components
A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�B⃗=B𝑥 �̂�+B𝑦 �̂�+B𝑧 �̂�
Let us have
t hen
A⃗ ∙ B⃗=A𝑥B𝑥+A𝑦 B𝑦+A𝑧 B𝑧
![Page 70: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/70.jpg)
Vector product
A⃗
B⃗θ
A⃗× B⃗=A B sin θ �̂�=C⃗
![Page 71: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/71.jpg)
Right hand rule
A⃗
B⃗
C⃗
θ
![Page 72: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/72.jpg)
Examples of vector product
τ⃗= r⃗× F⃗
= torque r = position F = force
L⃗=r⃗× p⃗L⃗=r p sin θ �̂�L = angular momentum r = position p = linear momentum
![Page 73: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/73.jpg)
Geometrical meaning of Vector product
θ
A
B
B sin θ θ
A
B
Asinθ
= Area of parallelogram made by two vectors
![Page 74: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/74.jpg)
Properties of Vector product
1The vector product is anti-commutative.
![Page 75: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/75.jpg)
Properties of Vector product
2The vector product is distributive over
addition.
A⃗× ( B⃗+C⃗ )=A⃗× B⃗+A⃗×C⃗
![Page 76: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/76.jpg)
Properties of Vector product
3The magnitude of the vector product of two
perpendicular vectors is maximum.
|A⃗× B⃗|=A B
![Page 77: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/77.jpg)
Properties of Vector product
4The vector product of two parallel vectors
is a null vector.
A⃗× B⃗=0⃗
![Page 78: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/78.jpg)
Properties of Vector product
5The vector product of two anti-parallel
vectors is a null vector.
A⃗× B⃗=0⃗
![Page 79: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/79.jpg)
Properties of Vector product
6The vector product of a vector with itself
is a null vector.
A⃗× A⃗=0⃗
![Page 80: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/80.jpg)
Properties of Vector product
7The vector product of two same unit
vectors is a null vector.
�̂�× �̂�= �̂�× �̂�=�̂�×�̂�¿ (1)(1)sin 0 ° �̂�=0⃗
![Page 81: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/81.jpg)
Properties of Vector product
8The vector product of two different unit
vectors is a third unit vector.�̂�× �̂�=�̂��̂�× �̂�=�̂��̂�× �̂�= �̂�
�̂�× �̂�=− �̂��̂�× �̂�=− �̂��̂�×�̂�=− �̂�
![Page 82: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/82.jpg)
Aid to memory
![Page 83: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/83.jpg)
Calculating vector product using components
A⃗=A 𝑥 �̂�+A 𝑦 �̂�+A𝑧 �̂�B⃗=B𝑥 �̂�+B𝑦 �̂�+B𝑧 �̂�
Let us have
t hen
![Page 84: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/84.jpg)
Calculating vector product using components
A⃗× B⃗=| �̂� �̂� �̂�A𝑥 A 𝑦 A𝑧
B𝑥 B𝑦 B𝑧|
![Page 85: 1 - Scalars & vectors](https://reader034.vdocuments.us/reader034/viewer/2022042504/58a2660b1a28abb92b8b645f/html5/thumbnails/85.jpg)
Thankyou