
Dot product of two vectorsPage Navigation:
Geometric interpretation. Dot product of two vectors a and b is a scalar quantity equal to the product of magnitudes of vectors multiplied by the cosine of the angle between vectors:
a · b = a · b cos α Algebraic interpretation. Dot product of two vectors a and b is a scalar quantity equal to the sum of pairwise products of coordinate vectors a and b.
Dot product is also call scalar product or inner product.
Dot product  formulasDot product formula for plane problemsIn the case of the plane problem the dot product of vectors a = {a_{x} ; a_{y}} and b = {b_{x} ; b_{y}} can be found by using the following formula: a · b = a_{x} · b_{x} + a_{y} · b_{y} Dot product formula for spatial problemsIn the case of the spatial problem the dot product of vectors a = {a_{x} ; a_{y} ; a_{z}} and b = {b_{x} ; b_{y} ; b_{z}} can be found by using the following formula: a · b = a_{x} · b_{x} + a_{y} · b_{y} + a_{z} · b_{z} Dot product formula for n dimensional space problemsIn the case of the n dimensional space problem the dot product of vectors a = {a_{1} ; a_{2} ; ... ; a_{n}} and b = {b_{1} ; b_{2} ; ... ; b_{n}} can be found by using the following formula: a · b = a_{1} · b_{1} + a_{2} · b_{2} + ... + a_{n} · b_{n} Properties of dot product of vectors
Dot product  exampleExamples of calculation of the dot product of vectors for plane problemsExample 1. Find the dot product of vectors a = {1; 2} and b = {4; 8}.
Solution: a · b = 1 · 4 + 2 · 8 = 4 + 16 = 20. Example 2. Find the dot product of vectors a and b, if their magnitudes is a = 3, b = 6, and the angle between the vectors is equal to 60˚.
Solution: a · b = a · b cos α = 3 · 6 · cos 60˚ = 9. Example 3. Find the dot product of vectors p = a + 3b and q = 5a  3 b, if their magnitudes is a = 3, b = 2, and the angle between the vectors a and b is equal to 60˚.
Solution: p · q = (a + 3b) · (5a  3b) = 5 a · a  3 a · b + 15 b · a  9 b · b == 5 a^{2} + 12 a · b  9 b^{2} = 5 · 3^{2} + 12 · 3 · 2 · cos 60˚  9 · 2^{2} = 45 +36 36 = 45. Examples of calculation of the dot product of vectors for spatial problemsExample 4. Find the dot product of vectors a = {1; 2; 5} and b = {4; 8; 1}.
Solution: a · b = 1 · 4 + 2 · 8 + (5) · 1 = 4 + 16  5 = 15. Examples of calculation of the dot product of vectors for n dimensional space problemsExample 5. Find the dot product of vectors a = {1; 2; 5; 2} and b = {4; 8; 1; 2}.
Solution: a · b = 1 · 4 + 2 · 8 + (5) · 1 + 2 · (2) = 4 + 16  5 4 = 11.
Vectors
Vectors Definition. Main information
Component form of a vector with initial point and terminal point
Length of a vector
Direction cosines of a vector
Equal vectors
Orthogonal vectors
Collinear vectors
Coplanar vectors
Angle between two vectors
Vector projection
Addition and subtraction of vectors
Scalarvector multiplication
Dot product of two vectors
Cross product of two vectors (vector product)
Scalar triple product (mixed product)
Linearly dependent and linearly independent vectors
Decomposition of the vector in the basis
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