Vector Flashcards
Why is current scalar if it has a magnitude and direction
If you imagine a current of 5A flowing from 2 wires each into a single wire , then the current in the final wire would be 10A. But according to vector algebra (addition) the current flowing in the final wire would be √50. So current is a scalar qty.
Why is time scalar ?
It is because time has only one direction, that is forwards. Time cannot go backwards as proven by Einstein’s theory (speed of light).
As time has only one direction.
By adding a bit to the definition of the vector quantity. ‘A vector quantity is a scalar with a direction in the three-dimensional space.’
Time is considered to be the 4th dimension
And Time does not follow vector algebra(addition)
Why is pressure scalar?
Pressure has both magnitude and direction but it always has only one direction, perpendicular to the surface it’s being applied to.
Polar vectors
Vectors having starting points
Axial vectors also called
Pseudovectors
Null vectors also called
0 vector
Null vector
0 magnitude and it’s direction is indeterminate
Null vector denoted by
→
0
Unit vector of →
A
A cap or A hat or A caret
A unit vector specifies
Only direction of vector A and no magnitude (no unit also)
The direction of x axis is represented by
i cap
The direction of y axis is represented by
j cap
The direction of z axis is represented by
k cap
Equal vectors have
Equal magnitude and same direction
Like vectors have
Unequal magnitude and same direction
Unlike vectors have
Unequal magnitude and opposite direction
Opposite vectors have
Equal magnitude and opposite direction
Unit vector =
Vector / Magnitude of the vector
Opposite vectors also called
Negative vectors
Coplanar vectors
Vectors lying in the same plane. E.g. i cap and j cap , j cap and k cap
Orthogonal vectors
Perpendicular vectors
If a vector is displayed parallel to itself :
Its value doesn’t change
E.g. of null vectors
When a body is moving with a constant velocity then the acceleration vector would be 0
3 law used for vector addition
Triangle law
Parallelogram law
Polygon law
Magnitude of a vector calculated using its coordinates
|A| = √( x² + y² + z² )
Sines of major angles
0° = 0 30° = 1/2 45° = 1/√2 60° = √3/2 90° = 1
Cosine of major angles
0° = 1 30° = √3/2 45° = 1/√2 60° = 1/2 90° =0
Coplanar vectors
Vectors lying in the same planes
Sum of 2 vectors =
√(a² + b² + 2ab cosθ)
Angle between 2 vectors =
Q sinθ / P+Q cosθ
1 radian definition and formula
Angle subtended at the centre of the circle by an arc equal to the radius of the circle
1°= how many minutes and seconds
1° = 60’ 1’ = 60” 1° = 60 x 60” = 3600”
Minutes and seconds of an angle are represented by
1 minute = 1’
1 second = 1”
Positive angle
If the amount of rotation between 2 lines is in the anti clockwise direction, the angle is considered to be +ve.
Negative angle
If the amount of rotation between 2 lines is in the clockwise direction, the angle is considered to be -ve.
Angle definition
The amount of rotation needed to get the terminal side from the initial side.
Triangle vector addition method also called
Tail - Head method. The tail of the vector is joined to the head of the next vector.
Why is area considered to be scalar
Find area of a square with side 6 cm using graph. So take 2 length vectors with length 6 cm as x-y axes.
Dipole
A pair of equally and opposite charged particles separated by a distance.
δ :
Lowercase Greek delta
What is a plane in physics
A “plane” is simply a flat (2-dimensional) surface. It could be a real or an imaginary surface. It’s a surface with no thickness or curvature in the third dimension. In theory it has no edges and so is infinite.
Why is current density a vector ?
It is defined as the product of charge density and velocity for any location in space…
J = ρv
Sine of the angle 37°
3/5
Sin of the angle 53°
4/5
Cos of the angle 37°
4/5
Cos of angle 57
3/5
Tan of 90 is
Infinity
Sin (90 - theta) =
Cos theta
A vector v→ can be represented in another form like
V (bold face)
Dipole moment vector direction
Direction is along the lines joining the charges from -ve to +ve . (Direction : -ve to +ve)
Axial vectors definition
Vectors that represent rotational effect
Orthogonal unit vectors
Set of mutually perpendicular unit vectors
Equal vectors are like vectors , but not all like vectors are not equal vectors as :
Like vectors may or may not be equal.
To describe position of a vector using position vector, we need to select a point called
Origin (O)
Position vectors are represented by
→
r
Angle between 2 vectors means
Smaller of the angle between the vectors when they are placed together.
In triangle method of vector addition, the vectors to be added are kept in the direction of
Tail to head
Direction of the resultant formula:
Tan α = b sinθ / (a + b cosθ)
In parallelogram method of vector addition , vectors are added in …….. method
Tail to tail
Polygon law of vector addition is similar to
It is similar to triangle method of vector addition but here , it is done repetitively.
Resolution of a vector means
Splitting up of a single vector in different directions .
Component vectors
Vectors which are formed by the resolution of a vector.
Resolution of a vector is just opposite to
Composition of vectors.
Maximum no of component vectors a vector can have
Infinity but for for sake of simplicity, it is resolved into 2 or 3 perpendicular vectors.
Right handed vs left handed Cartesian coordinate system.
In right-handed Cartesian system the positive x and y axes point right and up, and the negative z axis points forward.
In left- handed Cartesian system , it is same , but the positive z axis point forwards instead of the negative one.
Adding of perpendicular vectors [Easy]
|a| = √[a(x)² + a(y)² + a(z)²]
This is because as cos of 90 is 0
Resolution of vectors can be used for simplified ……… of vectors
Addition
If the multiplication of 2 vectors results in a scalar it is called
Scalar product or dot product.
Scalar product of 2 vectors is done by
A . B = |a| |b| cos θ
A.B - read as a dot b
The scalar product properties
Commutative
Distributive
We can find the angle between 2 vectors by scalar product
Cos θ = (a x b) / (|a| x |b|)
Power is defined as the dot product of
Force and velocity
If the multiplication of 2 vectors is a vector then it is called
Vector product
Vector product is also called as the
Cross product and is read as a cross b
The resultant of a vector product always points
Perpendicular to the plane formed by vectors a and b
Vector product formula
C = |a| x |b| x sin θ
Direction of the resultant of the cross product is determined by the
Right hand thumb rule
Vector product of 2 parallel vectors is
0 as the angle between them is 0 and as sin of 0 is 0.
Vector product properties
Not commutative as directions of resultant change.
E.g. A x B = -B x A
Vector product multiplication orders and signs
i x j = k
j x k = i
i x k = j
If the order is changed, then the resultant changes sign and becomes negative, that means the direction changes , that means the direction of the resultant becomes perpendicular in the downward direction.
Example of vectors that are results of vector products
Torque and angular momentum.
Torque is defined as the vector product of
The cross product of the position vector (radius) and the force vector.
τ = r x f
Angular momentum is represented by
L
Angular momentum can be defined as the vector product of
Position vector (radius) and linear momentum
L = r x p
If the multiplication of 2 vectors results in a scalar it is called
Scalar product or dot product.
Scalar product of 2 vectors is done by
A . B = |a| |b| cos θ
A.B - read as a dot b
The scalar product properties
Commutative
Distributive
E.g. of vectors formed by scalar products
Work and power
We can find the angle between 2 vectors by scalar product
Cos θ = (a x b) / (|a| x |b|)
Work is defined as the dot product of
Force and displacement
Power is defined as the dot product of
Force and velocity
If the multiplication of 2 vectors is a vector then it is called
Vector product
Vector product is also called as the
Cross product and is read as a cross b
The resultant of a vector product always points
Perpendicular to the plane formed by vectors a and b
Vector product formula
|a x b| = |a| x |b| x sin θ
Direction of the resultant of the cross product is determined by the
Right hand thumb rule
Vector product of 2 parallel vectors is
0 as the angle between them is 0 and as sin of 0 is 0.
Vector product properties
Not commutative as directions of resultant change.
E.g. A x B = -B x A
Vector product multiplication orders and signs
i x j = k
j x k = i
i x k = j
If the order is changed, then the resultant changes sign and becomes negative, that means the direction changes , that means the direction of the resultant becomes perpendicular in the downward direction.
Example of vectors that are results of vector products
Torque and angular momentum.
Torque is defined as the vector product of
The cross product of the position vector (radius) and the force vector.
τ = r x f
Angular momentum is represented by
L
Angular momentum can be defined as the vector product of
Position vector (radius) and linear momentum
L = r x p
