In this section we are going to understand the basic concept of vectors and we will see a very important operation that can be done on vectors which is cross product.
Let us see what are vectors first.Vectors are used to represent quantities where magnitude and direction both are essential to define a quantity properly. A vector hence represents magnitude and direction of a quantity. For example quantities like velocity and acceleration needs to be defined with magnitude and direction as well. A velocity of 5 km/hr due east represents that the object is moving with a speed of 5 km/hr in east direction. This means that velocity is a vector quantity.
A vector is denoted graphically by an arrow where the head of the arrow represents the direction of the vector and its length represents the magnitude of the vector. The length of a vector is used for comparing magnitude of 2 or more vectors. A vector with shorter length will have smaller magnitude than other vectors.
A vector a can be represented in 3 dimensional space as:
a= a1i+a2j+a3k
Here i, j and k are unit vectors in the x, y and z direction respectively.
Let us see how cross product can be performed on the vector Cross product is also known as vector product. This is due to the fact that the result after the product is also a vector. It is a binary operation on 2 vectors in 3 D space. The resulting vector is perpendicular to both the vectors and thus it is perpendicular to the plane in which initial vectors lie.
If the direction of the two vectors is same or they have zero magnitude or length, then the cross product of the two vectors results zero. The magnitude of the cross product vector is equal to the area of the parallelogram which has the two vectors as its sides. If the two vectors are perpendicular and form a rectangle, then the resulting magnitude will be the area of rectangle i.e. product of lengths of the vectors.
The operation of cross product is denoted by ×. If the two vectors are ‘a’ and ‘b’ then the cross product of the two vectors is denoted as a×b. Note that a×b is not equal to b×a. The resulting vector of this cross product denoted by c is a vector that is perpendicular to both the vectors. Its direction can be given by using right hand rule. Formula for Cross Product of Two Vectors is given as:
a×b=|a||b|sinӨ n
Here Ө is the smaller angle between the vectors a and b. |a| and |b| denotes the magnitude of the vectors a and b respectively. n here represents the unit vector perpendicular to the plane of a and b. Its direction can be given by right hand rule. The right hand rule says that if a×b is the cross product that you are calculating then role your fingers of right hand from vector a to vector b, the direction of your thumb will give the direction of resulting vector as shown below:
If the vectors a and b are parallel to each other, then the cross product of then is a zero vector as sin0 = 0
Cross product is considered as anti commutative. This means that a×b = -(b×a)
Let us see what are vectors first.Vectors are used to represent quantities where magnitude and direction both are essential to define a quantity properly. A vector hence represents magnitude and direction of a quantity. For example quantities like velocity and acceleration needs to be defined with magnitude and direction as well. A velocity of 5 km/hr due east represents that the object is moving with a speed of 5 km/hr in east direction. This means that velocity is a vector quantity.
A vector is denoted graphically by an arrow where the head of the arrow represents the direction of the vector and its length represents the magnitude of the vector. The length of a vector is used for comparing magnitude of 2 or more vectors. A vector with shorter length will have smaller magnitude than other vectors.
A vector a can be represented in 3 dimensional space as:
a= a1i+a2j+a3k
Here i, j and k are unit vectors in the x, y and z direction respectively.
Let us see how cross product can be performed on the vector Cross product is also known as vector product. This is due to the fact that the result after the product is also a vector. It is a binary operation on 2 vectors in 3 D space. The resulting vector is perpendicular to both the vectors and thus it is perpendicular to the plane in which initial vectors lie.
If the direction of the two vectors is same or they have zero magnitude or length, then the cross product of the two vectors results zero. The magnitude of the cross product vector is equal to the area of the parallelogram which has the two vectors as its sides. If the two vectors are perpendicular and form a rectangle, then the resulting magnitude will be the area of rectangle i.e. product of lengths of the vectors.
The operation of cross product is denoted by ×. If the two vectors are ‘a’ and ‘b’ then the cross product of the two vectors is denoted as a×b. Note that a×b is not equal to b×a. The resulting vector of this cross product denoted by c is a vector that is perpendicular to both the vectors. Its direction can be given by using right hand rule. Formula for Cross Product of Two Vectors is given as:
a×b=|a||b|sinӨ n
Here Ө is the smaller angle between the vectors a and b. |a| and |b| denotes the magnitude of the vectors a and b respectively. n here represents the unit vector perpendicular to the plane of a and b. Its direction can be given by right hand rule. The right hand rule says that if a×b is the cross product that you are calculating then role your fingers of right hand from vector a to vector b, the direction of your thumb will give the direction of resulting vector as shown below:
If the vectors a and b are parallel to each other, then the cross product of then is a zero vector as sin0 = 0
Cross product is considered as anti commutative. This means that a×b = -(b×a)
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