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Dot product of two vectors

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 = {ax ; ay} and b = {bx ; by} can be found by using the following formula:

a · b = ax · bx + ay · by

Dot product formula for spatial problems

In the case of the spatial problem the dot product of vectors a = {ax ; ay ; az} and b = {bx ; by ; bz} can be found by using the following formula:

a · b = ax · bx + ay · by + az · bz

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 ; ... ; an} and b = {b1 ; b2 ; ... ; bn} can be found by using the following formula:

a · b = a1 · b1 + a2 · b2 + ... + an · bn

Properties of dot product of vectors

  1. The dot product of a vector with itself is always greater than zero or equal to zero:

    a · a ≥ 0

  2. 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

  3. The dot product of a vector with itself is equal to the square of its magnitude:

    a · a = |a|2

  4. The dot product operation is communicative:

    a · b = b · a

  5. 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

  6. (αa) · b = α(a · b)
  7. 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.

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