Dot product of two vectors

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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 - formulas

Dot product formula for plane problems

In 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 problems

In 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 problems

In the case of the n dimensional space problem the dot product of vectors a = {a₁ ; a₂ ; ... ; a_n} and b = {b₁ ; b₂ ; ... ; b_n} can be found by using the following formula:

a · b = a₁ · b₁ + a₂ · b₂ + ... + a_n · b_n

Properties of dot product of vectors

The dot product of a vector with itself is always greater than zero or equal to zero:
a · a ≥ 0
The dot product of a vector with itself is zero if and only if the vector is the zero vector:
a · a = 0 <=> a = 0
The dot product of a vector with itself is equal to the square of its magnitude:
a · a = |a|²
The dot product operation is communicative:
a · b = b · a
If the dot product of two not zero vectors is is zero, then these vectors are orthogonal:
a ≠ 0, b ≠ 0, a · b = 0 <=> a ┴ b
(αa) · b = α(a · b)
The dot product operation is distributive:
(a + b) · c = a · c + b · c

Dot product - example

Examples of calculation of the dot product of vectors for plane problems

Example 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 · 3² + 12 · 3 · 2 · cos 60˚ - 9 · 2² = 45 +36 -36 = 45.

Examples of calculation of the dot product of vectors for spatial problems

Example 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 problems

Example 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 Scalar-vector 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

Online calculators with vectors

Tasks and exercises with vector 2D

Tasks and exercises with vector 3D

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