There are two ways to multiply vectors — both physically useful. The scalar (dot) product returns a number — a scalar. Used whenever "amount of projection" or "parallel component" matters — from work (force × distance) to flux (field × area).
Definition
For two vectors \vec a and \vec b with angle \phi between them:
\[ \boxed{\vec a \cdot \vec b = |a||b|\cos\phi} \]
Note: the result is a scalar (not a vector). It can be positive, negative, or zero.
Signs:
\phi < 90°: positive (vectors "same-ish direction")\phi = 90°: zero (vectors perpendicular)\phi > 90°: negative (vectors "opposing")
Why "dot"?
\vec a \cdot \vec b can be read geometrically: \vec a times the component of \vec b parallel to \vec a.
Component of \vec b along \vec a: b \cos\phi (the projection).
\[ \vec a \cdot \vec b = a \times (\text{projection of } \vec b \text{ onto } \vec a) \]
Component form
If \vec a = a_x \hat i + a_y \hat j + a_z \hat k and \vec b = b_x \hat i + b_y \hat j + b_z \hat k:
\[ \boxed{\vec a \cdot \vec b = a_x b_x + a_y b_y + a_z b_z} \]
Wonderfully simple. Multiply corresponding components and add. No angles involved — pure algebra.
Why this matches the original definition
Using unit-vector properties:
\hat i \cdot \hat i = 1(sincecos 0° = 1and each unit has magnitude 1)\hat i \cdot \hat j = 0(sincecos 90° = 0)
Expand \vec a \cdot \vec b:
\[ (a_x \hat i + a_y \hat j) \cdot (b_x \hat i + b_y \hat j) = a_x b_x (\hat i \cdot \hat i) + a_x b_y (\hat i \cdot \hat j) + a_y b_x (\hat j \cdot \hat i) + a_y b_y (\hat j \cdot \hat j) \]
\[ = a_x b_x + 0 + 0 + a_y b_y = a_x b_x + a_y b_y\ ✓ \]
Properties
1. Commutative: \[ \vec a \cdot \vec b = \vec b \cdot \vec a \]
2. Distributive: \[ \vec a \cdot (\vec b + \vec c) = \vec a \cdot \vec b + \vec a \cdot \vec c \]
3. Compatible with scalars: \[ (k \vec a) \cdot \vec b = k(\vec a \cdot \vec b) \]
4. Self dot product = magnitude squared: \[ \vec a \cdot \vec a = a^2 \]
Useful as a perpendicularity test: if \vec a \cdot \vec b = 0 and neither is zero, the two vectors are perpendicular.
Physical application 1 — work
If a force \vec F moves an object through displacement \vec d, the work done:
\[ W = \vec F \cdot \vec d = F d \cos\phi \]
Interpretation: only the force parallel to the displacement does work. The perpendicular component contributes zero.
Example: you pull a 100 kg load with a 500 N force at 30° above horizontal, over 10 m:
\[ W = 500 \cdot 10 \cdot \cos 30° = 5000 \cdot 0.866 \approx 4330\ \mathrm{J} \]
If the pull were fully horizontal (\phi = 0°): W = 5000 J. Pulling at an angle costs useful work.
Physical application 2 — angle between two vectors
From \vec a \cdot \vec b = |a||b|\cos\phi, solve for \phi:
\[ \cos\phi = \frac{\vec a \cdot \vec b}{|a||b|} = \frac{a_x b_x + a_y b_y + a_z b_z}{\sqrt{a_x^2 + a_y^2 + a_z^2} \sqrt{b_x^2 + b_y^2 + b_z^2}} \]
Example: \vec a = 3\hat i + 4\hat j, \vec b = -2\hat i + 1\hat j.
\[ \vec a \cdot \vec b = 3 \cdot (-2) + 4 \cdot 1 = -6 + 4 = -2 \]
\[ |a| = \sqrt{9 + 16} = 5,\quad |b| = \sqrt{4 + 1} = \sqrt{5} \]
\[ \cos\phi = \frac{-2}{5\sqrt{5}} \approx -0.179 \Rightarrow \phi \approx 100.3° \]
Perpendicularity test
If two vectors are perpendicular, their dot product is zero. Example: \vec a = 2\hat i + \hat j, \vec b = -\hat i + 2\hat j.
\[ \vec a \cdot \vec b = -2 + 2 = 0 \]
So they're perpendicular. A quick tool for orthogonality checks.
A few notes and common mistakes
1. The dot product doesn't yield a vector.
\vec a \cdot \vec b is a scalar. If you got a vector, you made a mistake.
2. Zero doesn't require zero vectors.
\vec a \cdot \vec b = 0 can mean: (a) \vec a = 0, (b) \vec b = 0, or (c) \vec a \perp \vec b. For nonzero vectors, it's (c).
3. Sign matters. Negative dot product means the angle is more than 90°. Positive means less than 90°.
4. For work, only the parallel component matters. Push a car horizontally and it moves horizontally — all the force does work. Push it perpendicular to its motion — no work.
What you should be able to do
After this section, you should be able to:
- Write the dot-product definition
|a||b|\cos\phi - Use the component form
a_x b_x + a_y b_y + a_z b_z - Compute mechanical work as
W = \vec F \cdot \vec d - Extract the angle between two vectors from components
- Perform the perpendicularity test
- Connect sign (
+,-,0) to angular relationship
Preview of §3.6
The second kind of product: the vector (cross) product. Unlike the dot product, this returns a vector — with direction set by the right-hand rule. Used for torque, angular momentum, Lorentz force on an electron. §3.6 is next.
📚 See also: Halliday Vol 1, Ch 3, §3.5 — Multiplying Vectors: Scalar Product.
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