Difficulty: Introductory to Intermediate | Prerequisites: vectors in R3, magnitude formula (Section 1.2 notes).
The dot product is the first way this course gives you to multiply two vectors together. Unlike vector addition, the result is a scalar (a single number), not another vector. It measures how much two vectors point in the same direction and is the tool you will reach for whenever you need an angle between vectors, a projection, or a test for perpendicularity. If you are comfortable computing magnitudes and working with vectors in component form, this section builds directly on that.
The dot product of two vectors gives a scalar that encodes the angle between them. You can compute it two ways: geometrically (magnitudes times cosine of the angle) or algebraically (multiply matching components and add). Projection vectors use the dot product to decompose one vector along the direction of another.
Zero vector
The vector with all components equal to zero: in R3, it is ⟨0, 0, 0⟩, notated as 0 with an arrow.
Think of it as a vector with no direction and no length. The dot product of any vector with the zero vector is 0.
Dot product (scalar product)
For two vectors v and w, the dot product is defined geometrically as v · w = ||v|| ||w|| cos(θ), where θ is the smallest angle between them. In component form: v · w = v₁w₁ + v₂w₂ + v₃w₃.
In simple terms, this is a single number that tells you how aligned two vectors are. Positive means they broadly point the same way, negative means they broadly point in opposite directions, and zero means they are perpendicular.
Orthogonal
Two vectors are orthogonal if the angle between them is 90°. Equivalently, their dot product equals 0.
Think of it as the mathematical word for "perpendicular."
Scalar component (comp)
The scalar component of v in the direction of w is comp_w(v) = (v · w) / ||w||. It is a signed scalar, not a vector.
In simple terms, this tells you "how far" v reaches in the direction of w. Positive if they share a direction, negative if they oppose.
Projection vector (proj)
The projection of v onto w is proj_w(v) = ((v · w) / ||w||²) w. It is a vector that lies along w.
Think of it as the shadow of v cast onto the line defined by w, if the light came straight down perpendicular to w.
v · w = ||v|| ||w|| cos(θ), where θ is the angle between the two vectors.
The result is a scalar, not a vector.
If either vector is the zero vector, the dot product is 0.
For v = ⟨v₁, v₂, v₃⟩ and w = ⟨w₁, w₂, w₃⟩: v · w = v₁w₁ + v₂w₂ + v₃w₃.
This is far faster to compute than the geometric definition, and it is how you will calculate dot products on exams.
Example: v = ⟨5, 8⟩ and w = ⟨7, −1⟩. Then v · w = 5(7) + 8(−1) = 35 − 8 = 27.
Rearrange the geometric formula: cos(θ) = (v · w) / (||v|| ||w||), so θ = cos⁻¹((v · w) / (||v|| ||w||)).
Step 1: compute v · w using the component formula.
Step 2: compute ||v|| and ||w||.
Step 3: divide and take the inverse cosine.
Example: v = ⟨3, −2, 5⟩ and w = ⟨−2, 4, 1⟩. v · w = 3(−2) + (−2)(4) + 5(1) = −6 − 8 + 5 = −9. ||v|| = √38, ||w|| = √21. cos(θ) = −9 / (√38 · √21). θ = cos⁻¹(−9 / √798) ≈ 1.895 radians ≈ 108.6°.
The negative dot product confirms the angle is obtuse (greater than 90°).
Commutative: v · w = w · v.
Self-dot: v · v = ||v||². This is a useful identity for simplifying expressions.
Distributive over addition: a · (v + w) = a · v + a · w.
Scalar factor: C(v · w) = (Cv) · w = v · (Cw).
Two vectors are orthogonal (perpendicular) if and only if v · w = 0.
This follows from cos(90°) = 0 in the geometric formula.
Quick check: if the dot product comes out to 0, the vectors are at right angles.
The scalar component of v in the direction of w: comp_w(v) = (v · w) / ||w||. This is a number (positive or negative), not a vector.
The projection vector of v onto w: proj_w(v) = ((v · w) / ||w||²) w. This is a vector that lies along w.
The projection formula has ||w||² in the denominator. A common error is to use ||w|| instead of ||w||².
The relationship: proj_w(v) = comp_w(v) × (w / ||w||). The scalar component gives the signed length, and multiplying by the unit vector in the w direction gives the vector.
\vec{v} \cdot \vec{w} = \|\vec{v}\| \, \|\vec{w}\| \cos(\theta)\vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_2 + v_3 w_3\theta = \cos^{-1}\!\left(\frac{\vec{v} \cdot \vec{w}}{\|\vec{v}\| \, \|\vec{w}\|}\right)\text{comp}_{\vec{w}}\,\vec{v} = \frac{\vec{v} \cdot \vec{w}}{\|\vec{w}\|}\text{proj}_{\vec{w}}\,\vec{v} = \frac{\vec{v} \cdot \vec{w}}{\|\vec{w}\|^2}\,\vec{w}The dot product is how physics computes work: work = force · displacement, where only the component of force in the direction of motion contributes. Projection vectors appear in computer graphics for lighting calculations, where you project a light-direction vector onto a surface normal to determine how brightly a surface is lit.
Students often confuse the dot product (a scalar) with the cross product (a vector, covered in Section 12.4). The dot product gives a number. The cross product gives a vector perpendicular to both inputs.
In the projection formula, the denominator is ||w||², not ||w||. Using ||w|| instead of ||w||² is one of the most common calculation errors on exams.
Students sometimes assume that if two vectors have the same magnitude, their dot product must be positive. This is wrong: direction matters. Two vectors of equal length pointing in opposite directions have a negative dot product.
Orthogonality means v · w = 0, but that does not mean the vectors have the same magnitude or any other special property beyond being perpendicular.
⚠️ "Find the angle between two vectors" is one of the most common exam questions in Calc III. The procedure is always the same: component dot product, then magnitudes, then inverse cosine.
⚠️ The projection formula is frequently tested. Memorise both the scalar component formula (comp) and the projection vector formula (proj), and know the difference between them.
⚠️ The orthogonality test (dot product = 0 implies perpendicular) appears in problems where you are asked to verify that two vectors are perpendicular, or to find a value of a parameter that makes them perpendicular.
⚠️ The four dot product properties (commutative, self-dot, distributive, scalar factor) are used to simplify expressions in later proofs and problems. Know them cold.
True or false: the dot product of two vectors is always a vector. (False. It is always a scalar.)
Fill in the blank: if v · w = 0, the vectors are ______. (Orthogonal, or perpendicular.)
True or false: v · v = ||v||. (False. v · v = ||v||², not ||v||.)
Fill in the blank: the denominator in the projection formula proj_w(v) is ______. (||w||².)
True or false: a negative dot product means the angle between the vectors is greater than 90°. (True.)
Q: Compute the dot product of v = ⟨2, −1, 3⟩ and w = ⟨4, 5, −2⟩.
A: v · w = 2(4) + (−1)(5) + 3(−2) = 8 − 5 − 6 = −3.
Q: Find the angle between v = ⟨1, 0, 1⟩ and w = ⟨0, 1, 1⟩.
A: v · w = 0 + 0 + 1 = 1. ||v|| = √2, ||w|| = √2. cos(θ) = 1 / (√2 · √2) = 1/2. θ = cos⁻¹(1/2) = π/3 radians (60°).
Q: Are the vectors ⟨3, −6, 2⟩ and ⟨4, 1, −3⟩ orthogonal?
A: Dot product = 3(4) + (−6)(1) + 2(−3) = 12 − 6 − 6 = 0. Yes, they are orthogonal.
Q: Find the scalar component of v = ⟨3, 4⟩ in the direction of w = ⟨1, 0⟩.
A: comp_w(v) = (v · w) / ||w|| = (3(1) + 4(0)) / 1 = 3. The scalar component is 3.
Q: Find the projection of v = ⟨6, 2⟩ onto w = ⟨3, 4⟩.
A: v · w = 18 + 8 = 26. ||w||² = 9 + 16 = 25. proj_w(v) = (26/25)⟨3, 4⟩ = ⟨78/25, 104/25⟩.
The dot product is the foundation for the cross product (Section 12.4), which gives a vector perpendicular to both inputs. Later, directional derivatives use the dot product of the gradient vector with a unit direction vector. In physics-based applications throughout the course, work and flux calculations depend on dot products and projections.
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