Vector product ( Multiplying vectors )





Vector Product:

If we multiply a vector with any other vector or scalar quantity then the result is called vector product.

 

Multiplying a vector by a scalar:

If we multiply a vector \overrightarrow{a} by a scalar K , then the result is a new vector.

The magnitude of the vector formed by multiplying \overrightarrow{a} and K , is the magnitude of \overrightarrow{a} multiplied by K and the direction of the vector is same as vector \overrightarrow{a} if K is positive and exactly oppsite of K is negative.

And to divide the vector \overrightarrow{a} with K we can simply multiply the vector \overrightarrow{a} with \frac{1}{k}

 

Multiplying a vector by a vector:

We can multiply a vector with another vector in two ways as follows:

Dot product or scalar product:

The dot product or scalar product of two vectors \overrightarrow{a} and \overrightarrow{b} is defined to be:

\overrightarrow{a} . \overrightarrow{b} = ab \cos \phi

Where , \phi is the angle between \overrightarrow{a} and \overrightarrow{b}.

There are actually two angles between \overrightarrow{a} and \overrightarrow{b} one \phi and another 360 - \phi we can use either angle because their cosine is the same.

The dot product or cross product of two vectors is a scalar and it is denoted by a dot in between two vectors , So the product is called dot product or scalar product.

 

If vector \overrightarrow{a} and \overrightarrow{b} are denoted in component form then their dot product can be calculated as :

\overrightarrow{a} . \overrightarrow{b} = (a_x \hat{i} + a_y \hat{j} + a_z \hat{k}) . (b_x \hat{i} + b_y \hat{j} + b_z \hat{k})

Thus , \overrightarrow{a} . \overrightarrow{b} = a_x b_x + a_y b_y +a_z b_z

 

Cross product or vector product:

The cross product or vector product of two vectors \overrightarrow{a} and \overrightarrow{b} is a third vector \overrightarrow{c} whose magnitude is given by:

\overrightarrow{c} = \overrightarrow{a} \times \overrightarrow{b} = ab \sin \phi

where, \phi is the smaller of the two angles between vector \overrightarrow{a} and \overrightarrow{b}.

And the direction of \overrightarrow{c} is perpendicular to both vectors \overrightarrow{a} and \overrightarrow{b}

To find the exact direction of the cross product of vectors you can use right hand rule of vector products which stats if you make the orientation of your right hand as shown the the figure below then the direction of your thumb is the direction of vector \overrightarrow{c} = \overrightarrow{a} \times \overrightarrow{b} :

vector product

vector product

 

 

The cross product or vector product is a vector quantity and is denoted by a cross between two vectors , so he product is called vector product or cross product.

Note that the order of vector multiplication is important because:

\overrightarrow{a} \times \overrightarrow{b} \not= \overrightarrow{b}\times \overrightarrow{a}

 

And if the vectors \overrightarrow{a} and \overrightarrow{b} are given in component form then their cross product can be calculated using the formula:

\overrightarrow{a} \times \overrightarrow{b} = (a_y b_z - b_y a_z) \hat{i}+ (a_z b_x - b_z a_x) \hat{j} + ( a_x b_y - b_x a_y) \hat{k}

Related posts:

  1. Adding vectors by components Adding vectors by components. Vector sum by component method....
  2. Unit Vector Unit Vector. What is unit vector? What are standard unit...
  3. Components of vectors Components of vectors , components of vectors in two and...
  4. Geometric addition of vectors Geometric addition of vectors , Adding vectors geometrically. Law of...
  5. Vectors and Scalars Vectors and Scalars. What is Vector quantity and scalar quantity?...