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How to calculate Dot Product of Two Vectors?

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Vector is a two-dimensional entity with both magnitude and direction. A vector can be seen geometrically as a directed line segment with an arrow denoting the direction and a length equal to the magnitude of the vector. The vector’s direction is from the tail to the head. If two vectors have the same magnitude and direction, they are the same. This indicates that if we take a vector and translate it to a new point (without rotating it), the vector we get at the end is the same as the one we started with.

Dot Product of Two Vectors

The product of the magnitudes of the two vectors and the cosine of the angle between the two vectors is called the dot product of vectors. The dot product of two vectors produces a resultant that is in the same plane as the two vectors. The dot product can be either a positive or negative real value. The dot product of two vectors a and b is depicted as:

a.b = |a||b|cosθ

Properties of Dot Product

  • Commutativity: \vec{a}.\vec{b}=\vec{b}.\vec{a}
  • Distributivity: \vec{a}.(\vec{b}+\vec{c})=\vec{a}.\vec{b}+\vec{a}.\vec{c}
  • (\vec a+\vec b). \vec c = \vec a. \vec c+ \vec b. \overrightarrow c
  • (\lambda\vec a) . \vec b=\lambda(\vec{a})\vec{b}
  • The square root of the dot product of the vector by itself is equal to the length of the vector, i.e., \vec a=\sqrt{\vec a . \vec a}.
  • (\overrightarrow a -\overrightarrow b). \overrightarrow c = \overrightarrow a. \overrightarrow c - \overrightarrow b. \overrightarrow c
  • \overrightarrow a. (\overrightarrow b - \overrightarrow c) = \overrightarrow a. \overrightarrow b - \overrightarrow a. \overrightarrow c

Sample Problems

Question 1. Two vectors with magnitudes 6 and 7 units have an angle of 60 degrees between them. Find their dot product.

Solution:

Given: |a| = 6 units, |b| = 7 units and θ = 60°

We know, dot product of two vectors = |a||b|cosθ

= 6 . (7) . cos(60°)

= 42(½)

⇒ a . b= 21

Question 2. Two vectors with magnitudes 10 and 5 units have an angle of 60 degrees between them. Find their dot product.

Solution:

Given: |a| = 10 units, |b| = 5 units and θ = 60°

We know, dot product of two vectors = |a||b|cosθ

= 10 . (5) . cos(60°)

= 50(½)

⇒ a . b= 25

Question 3. Two vectors with magnitudes 100 and 50 units have an angle of 60 degrees between them. Find their dot product.

Solution:

Given: |a| = 100 units, |b| = 50 units and θ = 60°

We know, dot product of two vectors = |a||b|cosθ

= 100 . (50) . cos(60°)

= 5000(½)

⇒ a . b= 2500

Question 4. Two vectors with magnitudes 4 and 2 units have an angle of 30 degrees between them. Find their dot product.

Solution:

Given: |a| = 4 units, |b| = 2 units and θ = 30°

We know, dot product of two vectors = |a||b|cosθ

= 4 . (2) . cos(30°)

= 8(√3/2)

⇒ a . b= 4√3

Question 5. Two vectors with magnitudes 6 and 8 units have an angle of 60 degrees between them. Find their dot product.

Solution:

Given: |a| = 6 units, |b| = 8 units and θ = 60°

We know, dot product of two vectors = |a||b|cosθ

= 6 . (8) . cos(60°)

= 48(1/2)

⇒ a . b= 24

Question 5. Two vectors with magnitudes 16 and 8 units have an angle of 60 degrees between them. Find their dot product.

Solution:

Given: |a| = 16 units, |b| = 8 units and θ = 60°

We know, dot product of two vectors = |a||b|cosθ

= 16 . (8) . cos(60°)

= 16. (8) . (1/2)

⇒ a . b= 64

Question 6. Two vectors with magnitudes 20 and 5 units have an angle of 60 degrees between them. Find their dot product.

Solution:

Given: |a| = 20 units, |b| = 5 units and θ = 60°

We know, dot product of two vectors = |a||b|cosθ

= 20 . (5) . cos(60°)

= 20 . (5) . (1/2)

⇒ a . b= 50

Question 7. Two vectors with magnitudes 30 and 5 units have an angle of 60 degrees between them. Find their dot product.

Solution:

Given: |a| = 30 units, |b| = 5 units and θ = 60°

We know, dot product of two vectors = |a||b|cosθ

= 30 . (5) . cos(60°)

= 15. (5) . 

⇒ a . b= 75



Last Updated : 16 Feb, 2022
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