Scalar product of Two vectors               DefinitionThe dot product of two vectors is one-dimensional concept. It is ava...
Scalar product of Two vectors      Geometrical InterpretationGiven the characteristics of the dot product of two vectors b...
If we have to take dot product of a unit vector π‘Žand 𝑏 is second vector of non-zero length,Then π‘Ž . 𝑏 is the length of vec...
𝑏                                πœƒ                                                                        π‘Ž               ...
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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‘Ž and 𝑏, if |π‘Ž| and |𝑏|represent the lengths of vectors π‘Ž and 𝑏, respectively, andif πœƒ is the angle between these vectors.Then, The dot product of vectors π‘Ž and 𝑏 will have the followingrelationship: π‘Ž. 𝑏= |π‘Ž||𝑏|cos πœƒ
  2. 2. Scalar product of Two vectors Geometrical InterpretationGiven the characteristics of the dot product of two vectors by therelation π‘Ž. 𝑏= |π‘Ž||𝑏|cos πœƒNow, we can interpret three possible conditions: 1. If π‘Ž and 𝑏 are perpendicular (at 90 degrees to each other), the result of the dot product will be zero, because cos πœƒ will be zero. 2. If the angle between π‘Ž and 𝑏 are less than 90 degrees, the dot product will be positive (greater than zero), as cos πœƒ will be positive, and the vector lengths are always positive values. 3. If the angle between π‘Ž and 𝑏 are greater than 90 degrees, the dot product will be negative (less than zero), as cos πœƒ 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‘Žand 𝑏 is second vector of non-zero length,Then π‘Ž . 𝑏 is the length of vector 𝑏 projected inthe direction of vector π‘Ž .That is, 𝑏 π‘Ž. 𝑏 = π‘Ž . 𝑏 cos πœƒ πœƒSince π‘Ž = 1, π‘Ž π‘Ž. 𝑏We have 𝑏 cos πœƒ π‘Ž. 𝑏 = 𝑏 cos πœƒ
  4. 4. 𝑏 πœƒ π‘Ž π‘Ž. 𝑏 𝑏 cos πœƒ . π‘ŽIf we have to take the dot product of π‘Ž andsecond vector 𝑏 of any non-zero length, then π‘Ž. 𝑏 is defined as (length of vector 𝑏 projected in the direction of unit vector vector of π‘Ž ) . π‘ŽThat is, π‘Ž. 𝑏 = 𝑏 cos πœƒ . π‘Ž

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