high school 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 and the sign goes from positive to 0 to negative; move the components and a·b is computed.

Geometric Meaning of Dot Product

Angle θ45°

👀 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

⑤On the first sketch |a|·|b| are fixed. Only θ moves

Dot Product via Components

a_x3

a_y1

b_x1

b_y3

🔢 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 = a₁ b₁ + a₂ b₂

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

④Projection is a formula box only. No sketch

Work It Out

Example 1

Given a=(1, 2), b=(3, −1), find a · b.

Substitute into the component formula a · b = a₁b₁ + a₂b₂.

a · b = 1·3 + 2·(−1)

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.

Find the dot product and the magnitudes of both vectors.

a · b = 1, |a| = √5, |b| = √10

Substitute into cosθ = (a · b)/(|a||b|) and simplify.

cosθ = 1√5·√10 = 1√50 = √210

▸ 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?

①−2

②1

③2

④3

⑤4

▸ ③ 2

If two vectors are perpendicular, then a · b = 0.

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)

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