Components and Dot Product of Plane Vectors Dot Product
The dot product is a single number telling how much two vectors point the same way.
Lean the same direction (acute angle) and it is positive, a right angle gives 0, and opening the opposite way (obtuse) makes it negative.
Geometrically it relates to the shadow (projection) one vector casts onto the other, yet you get the same value just by multiplying components and adding.
Turn the angle slider to watch the sign go from positive to 0 to negative, then move the components to see how a·b is computed.
Geometric Meaning of Dot Product
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Sign relationship between angle and dot product 👀 What is the Dot Product?
①A number indicating how 'aligned' two vectors are
②Green segment = projection of b onto a
③Acute (θ<90°): positive, right: 0, obtuse: negative
④Dot product = |a||b|cosθ
Dot Product via Components
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Components of two vectors and dot product result 🔢 Compute by Components
①For a = (a₁, a₂), b = (b₁, b₂)
②a · b = a₁b₁ + a₂b₂
③Multiply matching components and add
④This always equals |a||b|cosθ
Dot Product Formulas
Dot Product (Geometric)
a · b = | a || b | cosθ
Product of magnitudes and cosine of the angle
Dot Product (Component)
a · b = a 1 b 1 + a 2 b 2
Multiply by component and sum
Find the Angle
cosθ = a · b | a || b |
Dividing dot product by magnitudes gives cosθ
🔍 Linking the Two Formulas
①Component formula can be derived from the geometric definition
②Component formula is convenient in problem solving
③Use the cosθ formula to find the angle
④Orthogonality: a · b = 0 ↔ a ⊥ b
Projection and Applications
Projection Vector
proj a b = a · b | a |² a
Projection of b onto a
Projection Length
| proj a b | = | a · b | | a |
Magnitude of projection (ignoring sign)
💡 Meaning of Projection
①Projection = shadow of b dropped along a
②Physics: Work = Force · displacement = F · d cosθ = dot product
③Math: foot of perpendicular onto a line
Work It Out
Example 1
Given a=(1, 2), b=(3, −1), find a · b.
1 Substitute into the component formula a · b = a₁b₁ + a₂b₂.
a · b = 1·3 + 2·(−1)
2 Compute each term and add.
a · b = 3 − 2 = 1
▸ a · b = 1
Just multiply matching components and add — you can find the dot product without knowing the angle.
Example 2
Given a=(1, 2), b=(3, −1), find cosθ for the angle θ between them.
1 Find the dot product and the magnitudes of both vectors.
a · b = 1, |a| = √5, |b| = √10
2 Substitute into cosθ = (a · b)/(|a||b|) and simplify.
cosθ = 1 √5·√10 = 1 √50 = √2 10
▸ cosθ = √2/10
Find the dot product and the magnitudes separately, then plug into the cosθ formula to get the angle.
Summary
Orthogonality Condition
a ⊥ b ⟺ a · b = 0
Zero dot product means the vectors are perpendicular
2022 KICE mock exam Math (Geometry) type, adapted
If the vectors a=(2, 1) and b=(k, −4) are perpendicular, what is the value of k?
▸ ③ 2
1 If two vectors are perpendicular, then 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
①Two dot product formulas: |a||b|cosθ = a₁b₁ + a₂b₂
②Use cosθ to find the angle
③a · b = 0 ↔ perpendicular
④Projection vector and length formulas
⑤|a|² = a · a (magnitude squared = dot with itself)