high school Components and Dot Product of Plane Vectors
Dot Product
The dot product tells you, as a single number, how much two vectors point the same way.
It is positive when the angle between them is acute, 0 when it is a right angle and negative when it is obtuse.
Geometrically it is tied to the length of the projection of one vector onto the direction of the other, yet multiplying matching components and adding gives exactly the same value.
Turn the angle and the dot product goes from positive through 0 to negative; move the components to see how a · b is computed from them.
The Geometric Meaning of the Dot Product
seegongsik.com
Two vectors with |a| = 3 and |b| = 2.5, the angle between them and their dot product
45°
👀 What to read in the picture
①The red vector a has length 3 and the blue vector b has length 2.5; only the angle θ changes
②The green segment is the projection of b onto the direction of a, with signed length |b| cos θ
③Since a · b = |a||b| cos θ, the dot product is the length of a times the signed length of the green segment
④It is positive for θ < 90°, 0 for θ = 90° and negative for θ > 90°
The Dot Product from Components
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The components of two vectors, their dot product and the angle between them
3
1
1
3
🔢 Computing with components
①If a = (a₁, a₂) and b = (b₁, b₂)
②then a · b = a₁b₁ + a₂b₂
③Just multiply matching components and add
④This value always equals |a||b| cos θ, so you can work backward to the angle
Dot Product Formulas
Dot product (geometric)
a · b = | a || b | cosθ
The product of the two lengths and the cosine of the angle between them
Dot product (components)
a · b = a1 b1 + a2 b2
Multiply matching components and add
Finding the angle
cosθ = a · b| a || b |
Dividing the dot product by the product of the lengths gives cos θ
🔍 How the two formulas connect
①Using the law of cosines, the component formula can be derived from the geometric definition
②For calculation, the component formula is more convenient
③Find the angle with the cos θ formula
④Two nonzero vectors are perpendicular exactly when a · b = 0
Projection
Projection vector
proja b = a · b| a |² a
The projection of b onto the direction of a
Length of the projection
| proja b | = | a · b || a |
The length of the projection vector (the sign is dropped)
💡 Where projection is used
①The projection is the shadow of b dropped perpendicularly onto the line along a
②In physics, work is the dot product of force and displacement, F · d = |F||d| cos θ
③Projection also finds the foot of a perpendicular dropped onto a line
Worked Examples
Example 1
If a = (1, 2) and b = (3, −1), find a · b.
1
Substitute into the component formula a · b = a₁b₁ + a₂b₂.
a · b = 1 · 3 + 2 · (−1)
2
Work out each term and add.
a · b = 3 − 2 = 1
▸ a · b = 1
Multiplying matching components and adding gives the dot product without knowing the angle.
Example 2
If a = (1, 2) and b = (3, −1), find cos θ for the angle θ between them.
1
Find the dot product and the lengths of the two vectors.
a · b = 1, |a| = √5, |b| = √10
2
Substitute into cos θ = (a · b)/(|a||b|) and simplify.
cosθ = 1√5·√10 = 1√50 = √210
▸ cos θ = √2/10
Find the dot product and the lengths separately, then put them into the cos θ formula to get the angle.
Summary
Perpendicularity condition
a ⊥ b ⟺ a · b = 0
Two nonzero vectors are perpendicular when their dot product is 0
2022 KICE mock exam Math (Geometry) type, adapted
If the vectors a = (2, 1) and b = (k, −4) are perpendicular, what is k?
①−2
②1
③2
④3
⑤4
▸ ③ 2
1
Perpendicular vectors have a · b = 0.
2
Write the dot product in components, set it to 0 and solve for k.
a · b = 2k − 4 = 0 ⇒ k = 2
🎯 Exam Points
①The two forms of the dot product: |a||b| cos θ = a₁b₁ + a₂b₂
②Find cos θ to judge the angle
③a · b = 0 means perpendicular
④Keep the projection vector and the projection length formulas apart