Vectors Flashcards
Chapter 2
Scalars
A physical quantity that is completely described by its magnitude
-Temperature
-Speed
-Mass
-Volume
-Energy
Time
Vectors
Physical quantiles which have both magnitude and direction
-Velocity
-Displacement
-Weight
-Acceleration
Representation of a Vector
A Vector is represented by a line with an arrowhead. The point O from which the arrow starts is called the tail or initial point or origin of the vector. Point A where the arrow ends are called the tip or head or terminal point of the vector
Properties of Vectors
-Vectors are equal if they have the same magnitude and direction
-Vectors must have the same units for them to be added or subtracted
-The negative of a vector has the same magnitude but opposite direction
-Subtraction of a Vector is defined by adding negative vector
Magnitude of a Vector
𝑛-n-dimensional vector 𝑣 = (𝑣1, 𝑣2, …, 𝑣𝑛)
-Magnitude of vector OP = |OP|
____________
=/x^2 +y^2 +z^2
Unit Vector
A vector having unit magnitude is called a unit vector, a vector divided by its magnitude
Addition of Vectors
If two sides of a triangle are shown by continuous vectors (Vectors A and Vector B), then the third side of the triangle in the opposite direction shows the resultant of two vectors (Vector C)
-Vector addition is commutative A+B = B+A
-Vector Addition is associative
A + (B+C) = (A+B) +C
Subtraction of Vectors
When subtracting vectors, we change the direction of the vector to be subtracted and then add
Types of Vectors
-Negative Vector: Has the same magnitude but opposite direction of the given vector
-Equal Vector: If two vectors have equal magnitude and the same direction, then they are equal vectors
-Colinear Vectors: Two vectors acting along the same straight lines or parallel straight lines in the same direction or in the opposite direction
-Coplanar Vectors: When three or more vectors lie in the same plane
Zero Vectors: It is a vector with zero magnitude and no specific direction
Null Vector
When a vector is multiplied by zero, we get a vector whose magnitude is zero
-It has an arbitrary direction
-Is represented by a point
-Has zero magnitude
-Dot product of a null vector with a vector is zero
-Cross product of a null vector with any vector is also a null vector
-A null product is added or subtracted it gives back the given vector
Orthogonal Unit Vectors
The unit vectors along the X,Y,A axes are written as i,j,k respectively
Components of a Vector
The components of a vector in @D coordinate system are considered to be x- component and y-component
-V = =(Vx,Vy)-
-Horizontal Component Vx = Vcosθ
Vertical Component Vx = Vsinθ
Dot Product (Scalar Multiplication)
The product of the magnitude of the magnitude of one vector with the resolved component of the other in the direction of the first product
-Dot Product is commutative
ab = ba = abcosθ
-The dot product of a vector to itself is the magnitude squared of the vector
AA = AAcos0 = A^2
-The dot product of two mutually perpendicular vectors is zero
AB = ABcos90
-Dot product is distributive
A(B + C) = AB + AC
The product of two parallel vectors is equal to the product of their magnitudes
Cross Product (Vector Multiplication)
The cross product of two vectors, A and B, is denoted by AxB. The resultant vector is perpendicular to AxB
-Cross product is not commutative
AxB = BxA
Projection of a Vector
Projection of A on B(ProjBA)
AB
|B|
Projection of B on A(ProjAB)
BA
|A|
