Work, Energy and Power · Lesson 1 of 16
Scalar Product and Projections
“Build the vector mathematics needed to calculate work and power.”
• Interpret the dot product as a projection. • Calculate dot products from components. • Use the angle and unit-vector rules with correct units.
Why multiply two vectors?
A force can pull a suitcase sideways while the suitcase moves forward. Its effect on the energy transferred depends on how much of the force points along the displacement. Multiplying just the two magnitudes ignores this geometry. The scalar product includes the angle and produces one signed number, so it is well suited to quantities such as work. A scalar has no spatial direction; it can still be positive, zero or negative.
Projection gives the geometric meaning
Imagine dropping a perpendicular from the tip of B onto the line along A. The signed component along that line is B cosθ. Multiply this component by A to get the dot product. You can instead project A onto B: A cosθ multiplied by B gives the same result. A projection is negative when the vector points partly against the chosen positive direction. The projection is a component, with the units of the vector being projected; the dot product has the product of the two vectors’ units.
Signs and special angles
| Angle | Dot product | Meaning |
|---|---|---|
| 0° | +AB | Parallel in the same direction |
| 90° | 0 | Perpendicular |
| 180° | −AB | Opposite directions |
| Acute / obtuse | Positive / negative | Positive / negative projection |
Algebra and Cartesian components
The dot product is commutative because AB cosθ = BA cosθ. It is distributive over addition, so A·(B+C) = A·B + A·C. A scalar multiplier can be taken outside: A·(λB) = λ(A·B). For perpendicular Cartesian unit vectors, i-hat·i-hat = j-hat·j-hat = k-hat·k-hat = 1, while all mixed products are zero. Expanding the brackets therefore leaves only matching components.
Problem
F = (3, 4, −5) N and d = (5, 4, 3) m. Find their angle and the projection of F along d.
- 1.F·d = 3×5 + 4×4 − 5×3 = 16 N m = 16 J.
- 2.Both magnitudes are √50 in their respective units, so cosθ = 16/50 = 0.32 and θ ≈ 71.34°.
- 3.The signed scalar projection of F along d is F·d/|d| = 16/√50 ≈ 2.26 N. Its vector projection is (F·d/|d|²)d = (1.60, 1.28, 0.96) N.
- 4.The 16 J dot product and the 2.26 N projection are different physical quantities; checking units prevents confusion.
Problem
A body moves 4 m along +z under F = (−1, 2, 3) N. Calculate the work of this force.
- 1.The displacement is (0, 0, 4) m.
- 2.The dot product is −1×0 + 2×0 + 3×4 = 12 J.
- 3.The x and y force components may be balanced by constraints; they make no contribution to this force’s work along z.
Check your understanding
Quiz
A·B for perpendicular nonzero vectors is
Explanation: The projection along the other vector is zero.
Which quantity can be negative?
Explanation: An obtuse angle makes cosθ negative.
The projection of F along displacement has units of
Explanation: A force component remains a force.
For A = (2, −1, 0) and B = (3, 4, 5), A·B equals
Explanation: 2×3 − 1×4 = 2.
A·A equals
Explanation: The angle of a nonzero vector with itself is zero.
Key Takeaways
• The dot product measures alignment through a signed projection. • Expand matching components in a common coordinate system. • Dot products are scalars; their units are the product of the input units.
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Next · Lesson 2
Work by a Constant Force