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# Scalar product of vectors

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### Scalar product of vectors

1. 1. Scalar product of Two vectors DefinitionThe dot product of two vectors is one-dimensional concept. It is avalue expressing the angular relationship between the vectors.It is a scalar value as an operation of two vectors withthe same number of components.Lets say, we have two vectors, π and π, if |π| and |π|represent the lengths of vectors π and π, respectively, andif π is the angle between these vectors.Then, The dot product of vectors π and π will have the followingrelationship: π. π= |π||π|cos π
2. 2. Scalar product of Two vectors Geometrical InterpretationGiven the characteristics of the dot product of two vectors by therelation π. π= |π||π|cos πNow, we can interpret three possible conditions: 1. If π and π are perpendicular (at 90 degrees to each other), the result of the dot product will be zero, because cos π will be zero. 2. If the angle between π and π are less than 90 degrees, the dot product will be positive (greater than zero), as cos π will be positive, and the vector lengths are always positive values. 3. If the angle between π and π are greater than 90 degrees, the dot product will be negative (less than zero), as cos π will be negative, and the vector lengths are always positive values.
3. 3. If we have to take dot product of a unit vector πand π is second vector of non-zero length,Then π . π is the length of vector π projected inthe direction of vector π .That is, π π. π = π . π cos π πSince π = 1, π π. πWe have π cos π π. π = π cos π
4. 4. π π π π. π π cos π . πIf we have to take the dot product of π andsecond vector π of any non-zero length, then π. π is defined as (length of vector π projected in the direction of unit vector vector of π ) . πThat is, π. π = π cos π . π