Lecture 3: Applications of the Dot Product — Notes

Dot product recap and basic definitions

• Recall from last lecture: the dot product of two vectors measures how aligned they are and relates to the angle between them.

• In R^n for vector a = (a1, a2, …, an) and vector b = (b1, b2, …, bn), the dot product is:

  a dot b = sum from i=1 to n of (ai * bi)

• Geometric interpretation: a dot b = (magnitude of a) * (magnitude of b) * cos(theta) where theta is the angle between a and b.

• Magnitude:

  magnitude of a = square root of (a dot a)

• Quick relationships:

  • If theta = 0 degrees, then a dot b = (magnitude of a) * (magnitude of b)

  • If theta = 90 degrees, then a dot b = 0

• Useful scalar products: projection, work, etc.

Projection of a vector onto another vector

• Vector projection of vector a onto vector b (the component of a in the direction of b):

  proj_b(a) = ((a dot b) / (b dot b)) * b

• Scalar projection (component of a along b):

  comp_b(a) = (a dot b) / (magnitude of b) = (a dot b) / square root of (b dot b)

• Unit vector in the direction of a:

  unit vector a = a / (magnitude of a)

• Example (to mirror the transcript's setup):- Take vector a = (1,1,1) and vector b = (1,-3,1).

  • Compute: a dot b = 1(1) + 1(-3) + 1(1) = -1; b dot b = 1^2 + (-3)^2 + 1^2 = 11.

  • Vector projection:

    proj_b(a) = (-1/11) * (1, -3, 1) = (-1/11, 3/11, -1/11)

  • This projection lies in the direction of b and has magnitude equal to the scalar projection along b.

Sanity checks for projection

• The projection vector is parallel to b.

• The scalar projection is the length of the shadow of a on the line in the direction of b.

Work done by a force (dot product application)

• Definition in 3D: if a force F = (Fx, Fy, Fz) acts while the displacement is d = (dx, dy, dz), the work is:

  W = F dot d = Fxdx + Fydy + Fz*dz

• This is consistent with the scalar form W = (magnitude of F) * (magnitude of d) * cos(theta), where theta is the angle between F and d.

• Example 1 (incline example from transcript):- Force: F = <30, 0, 30> N

  • Displacement: d = 10 * m = <10cos(30 degrees), 0, 10sin(30 degrees)>

  • Compute components: d approx <8.6603, 0, 5>

  • Work:

    W = <30,0,30> dot <8.6603, 0, 5> = 30(8.6603) + 0(0) + 30(5) approx 259.8 + 150 = 409.8 J

  • Result: approximately 4.10 x 10^2 J (as it was in transcript, this might be a formatting error as it has no meaning in this context).

• Example 2 (rope-and-sled setup, as in transcript):- General relation: the work done by a force along the displacement is

  W = F dot d = Fxdx + Fydy + Fz*dz

  • If the rope makes an angle theta with the horizontal and the rope length is L, one common expression (under a simple setup) is

    W = F * L * cos(theta)

  • The transcript’s specific numeric setup is unclear for a definitive numeric answer, so the general form above is offered for clarity.

Lines in R^2

• A line in the plane can be written in implicit form as

  ax + by = c

  where the normal vector is n = (a, b).

• A direction vector along the line is

  u = (-b, a)

• If a point P = (x0, y0) lies on the line, then the line can be parameterized as

  (x, y) = (x0, y0) + t*(-b, a), where t is a real number.

• Equivalently, the line passes through P and Q, with Q-P parallel to u, so

  Q = P + t*(-b, a)

• Summary relation: for any point on the line, ax + by = c with c = ax0 + by0

Lines in R^3 (basic idea)

• A line in 3D can be written parametrically as

  r(t) = P + t*v

  where P = (x0, y0, z0) is a point on the line and v = (v1, v2, v3) is a direction vector.

• In coordinates: r(t) = (x0 + v1t, y0 + v2t, z0 + v3*t).

Planes in R^3

• A plane with normal vector n = (a, b, c) passing through a point P = (x0, y0, z0) satisfies n dot (Q - P) = 0 or equivalently a(x - x0) + b(y - y0) + c(z - z0) = 0

• In standard form: ax + by + cz = d, with d = ax0 + by0 + c*z0

• Key geometric fact: two planes are either parallel (same or opposite normals up to a scale) or they intersect in a line.

Intersection line of two planes (example from transcript)

• Given two planes:

  • S1: a1x + b1y + c1*z = d1

  • S2: a2x + b2y + c2*z = d2

• Solve the system to obtain a line of intersection. A common approach is to parametrize by setting one variable and solving for the other two.

• Transcript example (interpreted):- S1: x + y + 2z = 3

  • S2: x + 2y + 3z = 6

  • Two points on the line:

  • Set z = 0: x + y = 3; x + 2y = 6 => y = 3, x = 0 => point A = (0, 3, 0).

  • Set y = 0: x + 2z = 3; x + 3z = 6 => z = 3, x = -3 => point B = (-3, 0, 3).

  • Direction vector: d = B - A = (-3, 0, 3) - (0, 3, 0) = (-3, -3, 3) = -3*(1, 1, -1) (can simplify to (-1, -1, 1)).

  • Parametric form of the intersection line:

    (x, y, z) = (0, 3, 0) + t*(-1, -1, 1), where t is a real number.

• Cartesian form can be obtained from solving the system as shown above.

Angle between two planes

• The angle between planes is the angle between their normal vectors. If the normals are n1 and n2, then

  cos(theta) = |n1 dot n2| / ((magnitude of n1) * (magnitude of n2))

• Transcript example (using the interpreted normals):- For S1: x + y + 2z = 3 and S2: x + 2y + 3z = 6, the normals are n1 = (1, 1, 2) and n2 = (1, 2, 3).

  • Dot product: n1 dot n2 = 11 + 12 + 2*3 = 9.

  • Norms: magnitude of n1 = sqrt(1^2 + 1^2 + 2^2) = sqrt(6) and magnitude of n2 = sqrt(1^2 + 2^2 + 3^2) = sqrt(14).

  • Therefore

    cos(theta) = 9 / (sqrt(6) * sqrt(14)) = 9 / sqrt(84) approx 0.983

    so the angle is approximately

    theta approx arccos(0.983) approx 11 degrees

• Note on alternate normals (as seen in the transcript): if different normals are used, e.g., n1 = (1,1,1) and n2 = (1,2,3), then

  cos(theta) = |n1 dot n2| / ((magnitude of n1) * (magnitude of n2)) = 6 / (sqrt(3) * sqrt(14)) = 6 / sqrt(42) approx 0.926

  giving theta approx 22 degrees .

• Takeaway: the angle depends on the actual normals of the two planes; consistent normals are essential for a precise angle.

Connections and implications

• Practical use: dot products and projections are fundamental in decomposing forces, determining components along directions, and understanding work done by forces.

• Geometric interpretation: lines and planes in 2D and 3D can be analyzed using dot products, normals, and direction vectors.

• Real-world relevance: work calculations for inclined surfaces, pulling with ropes, and determining angles between planes in engineering and physics.

Quick reference formulas (summary)

• Dot product: a dot b = sum over i of ai * bi

• Magnitude: magnitude of a = sqrt(a dot a)

• Projection onto b: proj_b(a) = ((a dot b) / (b dot b)) * b

• Scalar projection: comp_b(a) = (a dot b) / (magnitude of b)

• Unit vector in direction of a: unit vector a = a / (magnitude of a)

• Work: W = F dot d

• Line in R^2: ax + by = c, u = (-b, a), (x,y) = (x0,y0) + t*(-b,a)

• Line in R^3: r(t) = P + t*v

• Plane in R^3: ax + by + cz = d, n=(a,b,c)

• Intersection line of two planes: solve the system of two linear equations; parametrize to obtain a line.

• Angle between planes: cos(theta) = |n1 dot n2| / ((magnitude of n1) * (magnitude of n2))