Level 6 · Ages 16–17
Read it again any time, including while you work through the exercises.
You know how to add vectors. But can you multiply them? Yes, and the result is surprising: the dot product of two vectors is not a vector but a plain number (a scalar). This number encodes precious information: how much the two vectors point the same way.
Two formulas to compute it, depending on the data. With coordinates, in an orthonormal system: if u has coordinates (x, y) and v has (x', y'), then u · v = xx' + yy'. Simple: product of the x's plus product of the y's. With norms and angle: u · v = ||u|| × ||v|| × cos(θ), where θ is the angle between the vectors.
The sign of the dot product tells a story. Positive? The angle is acute, the vectors roughly agree. Negative? The angle is obtuse, they mostly pull opposite ways. Zero? The most famous case: the vectors are orthogonal (perpendicular). The test u · v = 0 is THE royal method to prove a right angle in a coordinate system.
Where does that cos(θ) come from? Think of a force pulling a cart: only the part of the force aligned with the motion actually works. The dot product measures exactly that efficiency: that is why physicists love it.
On the menu: computations with both formulas, the orthogonality test, and finding an unknown coordinate to make two vectors perpendicular.
In an orthonormal system, u(2, 3) and v(4, -1). Compute u · v.
u · v = xx' + yy'
I have coordinates: I use the formula product of x's plus product of y's.
2 × 4 + 3 × (-1) = 8 - 3
I compute each product: 2 times 4 is 8, and 3 times -1 is -3. Watch the sign of the second product!
u · v = 5
The sum gives 5. Positive: the angle between u and v is acute, the vectors roughly point the same way.
Result: u · v = 5.
||u|| = 3, ||v|| = 4 and the angle between u and v is 60°. Compute u · v.
u · v = ||u|| × ||v|| × cos(60°)
No coordinates here, but norms and the angle: I use the cosine formula.
3 × 4 × 0,5 = 6
cos(60°) = 0.5, so u · v = 3 × 4 × 0.5 = 6. Positive, as expected for an acute 60° angle.
Result: u · v = 6.