Properties and orthogonal projection

Properties you need to know by heart

The scalar product behaves almost like ordinary multiplication:

• Symmetry: u.v = v.u
• Bilinearity: u·(v + w) = u·v + u·w and (k·u)·v = k × (u·v)
• u·u = ||u||² (the dot product of a vector with itself is the square of its norm)
• If u and v are orthogonal, u·v = 0 (and conversely)

The orthogonal projection: a very effective method

To calculate AB·AC, you can project C orthogonally onto the line (AB). If H is this projection, then:

AB·AC = AB·AH (since AB·HC = 0, as HC is perpendicular to AB)

And AB·AH can be easily calculated depending on whether H is on the same side as B or on the opposite side:

AB·AH = AB × AH if H and B are on the same side of A AB·AH = –AB × AH otherwise

Example

In triangle ABC, the foot of the height drawn from C is H, with AH = 3 and AB = 7 (H lies between A and B). Therefore, AB·AC = AB·AH = 7 × 3 = 21.

A very useful special case: the rectangle

In a rectangle ABCD, AB·AD = 0 because the sides are perpendicular. This allows us to simplify many dot product calculations by decomposing a vector into two perpendicular components.

Common pitfall

The orthogonal projection gives a NEGATIVE scalar product if the foot of the height lies outside the segment, on the opposite side. Always draw a diagram to check the direction of the vectors before multiplying the lengths.