dot product algebra and geometric applications

(Example)

Dot-Product Algebra and Geometric Applications

The dot product is bilinear:

u ⋅ (v + w ) = u ⋅ v + u ⋅ w,

(au ) ⋅ (bv ) = ab(u ⋅ v).

In Cartesian components,

|--------------------------| u ⋅ v = uxvx + uyvy + uzvz.| (1) ----------------------------

Example 1

For u = (2,−1, 3) and v = (0, 2, 4),

u ⋅ v = 10.

Hence

10 cos 𝜃 = √---√---- 14 20

and 𝜃 ≈ 53.3^∘.

Example 2: angle in a semicircle

If AB is a diameter of a circle and P lies on the circle, a vector proof gives

−→ −−→ AP ⋅BP = 0,

so the inscribed angle APB is a right angle.

Figure 15a, modernized: dot-product proof of Thales’ theorem.

Example 3: law of cosines

For triangle ABC,

a = b − c

implies

2 2 2 a = b + c − 2bc cosA.

Figure 15b, modernized: vector derivation of the law of cosines.

Example 4: distance from a point to a line

For A = (3, 1,−1) and the line through B = (2, 3, 0), C = (−1, 2, 4),

−→ −−→ AB = (− 1, 2,1), BC = (− 3,− 1,4).

The scalar projection of − →
AB along the line is

−A→B ⋅ −B−C→ 5 --------- = √----. ∥−−B→C ∥ 26

Therefore the perpendicular distance is

∘ ------------------- ( )2 ∘ ---- d = ∥−A→B ∥2 − √-5-- = 131-≈ 2.245. 26 26

Example 5: equation of a plane

A plane through P₁ = (x₁,y₁,z₁) with Normal n = (A,B,C) satisfies

|----------------| |n ⋅ (r − r ) = 0, ----------1------

A (x − x1) + B(y − y1) + C (z − z1) = 0.

Thus for a plane

Ax + By + Cz + D = 0,

n = (A,B,C) is a normal vector.

Example 6: point-to-plane distance

The distance from P₁ = (x₁,y₁,z₁) to Ax + By + Cz + D = 0 is

|----------------------------| | |Ax1 + By1 + Cz1 + D | | |d = ----√--2-----2----2----.| (2) -----------A--+-B--+-C--------

Source problems

Find the scalar component of (2, 3,−1) along the direction (−1,−2, 2).
Find the component of 2î + 3ĵ + k along a line in the first octant making equal angles with the coordinate axes.
Find the component of î + 2ĵ −k along the direction î −ĵ + k.
Find the angles between: (a) (1, 1, 0) and (1, 0, 1); (b) (1, 1, 1) and (1, 0, 0).
Find the shortest distance from A = (2,−3,−4) to the line through B = (1, 2,−3) and C = (3, 3,−5).
Find the shortest distance from A = (1,−2, 1) to 4x − 3y + 12z − 8 = 0.
Find the plane perpendicular to the line through A = (3, 4,−1) and B = (5, 2, 3) at its midpoint.
Find the angle between x − y + z + 2 = 0 and 2x + y − z + 1 = 0.
Prove the parallelogram law $∥a + b∥2 + ∥a − b ∥2 = 2∥a∥2 + 2∥b ∥2.$
Show that the sphere with diameter endpoints having position vectors a,b is described by $(r − a) ⋅ (r − b) = 0.$

Modern notation references

The notation and terminology in this modernized article follow standard present-day mechanics and vector-analysis usage, particularly:

J. R. Taylor, Classical Mechanics, University Science Books, 2005.
D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison–Wesley, 2002.

Source

This article is a modernized restatement of the corresponding Public Domain article in Louis Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.” The original 1930 edition is the source basis.

"dot product algebra and geometric applications" is owned by bloftin.
(view preamble)

dot product algebra and geometric applications