Question and Answer forum for K12 Students
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
The division of vector A by a non-zero scalar m is defined as the multiplication of A by \(\frac{1}{m}\). In this, the unit of resultant vector is the m division of unit A and m.
Scalars and Vectors Topics:
Scalar Product of Two Vectors a and b is:
The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them. It is denoted by (dot).
A • B = AB cosθ
The scalar or dot product of two vectors is a scalar.
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Scalar or Dot Product Properties
(i) Scalar product is commutative,
i.e. A • B = B • A
(ii) Scalar product is distributive,
i.e. A • (B+ C) = A • B + A • C
(iii) Scalar product of two perpendicular vectors is zero.
A • B = AB cos 9o° = 0
(iv) Scalar product of two parallel vectors or anti-parallel vectors is equal to the product of their magnitudes, i.e.
A • B = AB cos o° = AB (for parallel)
A • B = AB cos 180° = -AB (for anti-parallel)
(v) Scalar product of a vector with itself is equal to the square of its magnitude, i.e.
A • A = AA cos 0° = A²
(vi) Scalar product of orthogonal unit vectors
\(\begin{array}{l}
\hat{\mathbf{i}} \cdot \hat{\mathbf{i}}=\hat{\mathbf{j}} \cdot \hat{\mathbf{j}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{k}}=1 \\
\hat{\mathbf{i}} \cdot \hat{\mathbf{j}}=\hat{\mathbf{j}} \cdot \hat{\mathbf{k}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{i}}=0
\end{array}\)
(vii) Scalar product in cartesian coordinates
\(\mathbf{A} \cdot \mathbf{B}=\left(A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}\right) \cdot\left(B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}}\right)\)
= AxBx + AyBy + AzBz
Scalars and Vectors Topics:
Vector Product of Two Vectors a and b is:
The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by x (cross).
A x B = AB sin θ \(\hat{\mathbf{n}}\)
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Vector Cross Product Properties
(i) Vector product is not commutative, i.e.
A x B ≠ B x A [∴ (A x B) = – (B x A)]
(ii) Vector product is distributive, i.e.
A x (B + C)= A x B + A x C
(iii) Vector product of two parallel vectors is zero, i.e.
A x B = AB sin 0° = 0
(iv) Vector product of any vector with itself is zero.
A x A = AA sin 0° = 0
(v) Vector product of orthogonal unit vectors
\(\hat{\mathbf{i}} \times \hat{\mathbf{i}}=\hat{\mathbf{j}} \times \hat{\mathbf{j}}=\hat{\mathbf{k}} \times \hat{\mathbf{k}}=0\)
and \(\hat{\mathbf{i}} \times \hat{\mathbf{j}}=-\hat{\mathbf{j}} \times \hat{\mathbf{i}}=\hat{\mathbf{k}}\)
\(\hat{\mathbf{j}} \times \hat{\mathbf{k}}=-\hat{\mathbf{k}} \times \hat{\mathbf{j}}=\hat{\mathbf{i}}\)
\(\hat{\mathbf{k}} \times \hat{\mathbf{i}}=-\hat{\mathbf{i}} \times \hat{\mathbf{k}}=\hat{\mathbf{j}}\)
(vi) Vector product in cartesian coordinates
\(\mathbf{A} \times \mathbf{B}=\left(A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}\right) \times\left(B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}}\right)\)
= (AyBy – AzBy)\(\hat{\mathbf{i}}\) – (AxBz – BxAz)\(\hat{\mathbf{j}}\) + (AxBy – AyBx)\(\hat{\mathbf{k}}\)
Direction of Vector Cross Product
When C = A x B, the direction of C is at right angles to the plane containing the vectors A and B. The direction is determined by the right hand screw rule and right hand thumb rule.
(i) Right Hand Screw Rule:
Rotate a right handed screw from first vector (A) towards second vector (B). The direction in which the right handed screw moves gives the direction of vector (C).
(ii) Right Hand Thumb Rule:
Curl the fingers of your right hand from A to B. Then, the direction of the erect thumb will point in the direction of A x B.
Scalars and Vectors Topics:
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Multiplication of a Vector:
1. Multiplication of a Vector By a Real Number Definition:
When a vector A is multiplied by a real number n, then its magnitude becomes n times but direction and unit remains unchanged.
2. Multiplication of a Vector By a Scalar Definition:
When a vector A is multiplied by a scalar S, then its magnitude becomes S times and unit is the product of units of A and S but direction remains same as that of vector A.
Scalars and Vectors Topics:
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Subtraction of Vectors
Subtraction of a vector B from a vector A is defined as the addition of vector –B (negative of vector B) to vector A. Thus, A – B = A + (-B)
Scalars and Vectors Topics:
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Direction Cosines of a Vector:
If any vector A subtend angles α, β and γ with X-axis, Y-axis and Z-axis respectively and its components along these axes are Ax, Ay and Az, then
cos α = \(\frac{A_{x}}{A}\)
cos β = \(\frac{A_{y}}{A}\)
cos γ = \(\frac{A_{z}}{A}\)
Then, cos² α + cos² β + cos² γ = 1
Scalars and Vectors Topics:
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Rotation of a Vector By an Angle:
(i) If a vector is rotated through an angle θ, which is not an integral multiple of 2π, the vector changes.
(ii) If the frame of reference is rotated or translated, the given vector does not change. The components of the vector may, however, change.
Resolution of Vectors into Two Components
If two-component vectors of R are OP and PQ in the direction of A and B, respectively, and suppose OP = λA and PQ = µB, where λ and µ are two real numbers.
Then, resultant vector, R = λA + µB
Resolution of a Vector into its Rectangular Components
If any vector A subtends an angle θ with X-axis, then its
horizontal component, Ax = A cos θ
Vertical component, Ay = A sin θ
Magnitude of vector, A = \(\sqrt{A_{x}^{2}+A_{y}^{2}}\)
tan θ = \(\frac{A_{y}}{A_{x}}\)
Angle, θ = tan-1\(\left(\frac{A_{y}}{A_{x}}\right)\)
Scalars and Vectors Topics:
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
1. Triangle Law of Vectors Addition:
If two vectors acting at a point are represented in magnitude and direction by the two sides of a triangle taken in one order, then their resultant is represented by the third side of the triangle taken in the opposite order.
If two vectors A and B acting at a point are inclined at an angle 0, then their resultant
R = \(\sqrt{A^{2}+B^{2}+2 A B \cos \theta}\)
If the resultant vector R subtends an angle β with vector A, then
tan β = \(\frac{B \sin \theta}{A+B \cos \theta}\)
2. Parallelogram Law of Vectors Addition:
If two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram draw from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point.
Resultant of vectors A and B is given by
R = \(\sqrt{A^{2}+B^{2}+2 A B \cos \theta}\)
If the resultant vector R subtends an angle β with vector A, then
tan β = \(\frac{B \sin \theta}{A+B \cos \theta}\)
3. Polygon Law of Vectors Addition:
It states that, if number of vectors acting on a particle at a time are represented in magnitude and direction by the various sides of an open polygon taken in same order, then their resultant vector is represented in magnitude and direction by the closing side of polygon taken in opposite order. In fact, polygon law of vectors is the outcome of triangle law of vectors.
R = A + B + C + D + E
OE = OA + AB + BC + CD + DE
Properties of Vector Addition
(i) Vector addition is commutative, i.e. A + B = B + A
(ii) Vector addition is associative, i.e. A + (B + C) = B + (C + A) = C + (A + B)
(iii) Vector addition is distributive, i.e. m (A + B) = mA + mB
(iv) A + 0 = A
Scalars and Vectors Topics:
Tensors
Tensors are those physical quantities which have different values in different directions at the same point.
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Moment of inertia, radius of gyration, modulus of elasticity, pressure, stress, conductivity, resistivity, refractive index, wave velocity and density etc are the examples of tensors. Magnitude of tensor is not unique.
Scalars and Vectors Topics:
Scalars:
Those physical quantities which require only magnitude but no direction for their complete representation are called scalars.
Distance, speed, work, mass, density etc are the examples of scalars. Scalars can be added, subtracted, multiplied or divided by simple algebraic laws.
We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.
Vectors:
Those physical quantities which require magnitude as well as direction for their complete representation and follows vector laws are called vectors.
Vectors can be mainly classified into following two types
1. Polar Vectors:
These vectors have a starting point or a point of application such as displacement, force etc.
2. Axial Vectors:
These vectors represent rotational effect and act along the axis of rotation in accordance with right hand screw rule, such as angular velocity, torque, angular momentum etc.
Scalars and Vectors Topics: