Emath lecture recording on 25 February 2025 at 08.54.06 AM

Shortest Distance Calculation

• To find the shortest distance from a point x to a line with direction vector d:

  • Calculate the vector from the point to the line: x - Projection of x onto d

  • Find the length of that resulting vector.

Orthogonality of Vectors

• If vectors x and d are orthogonal:

  • Projection of x onto d = Zero vector.

Cross Product Introduction

• The cross product is a method used to find a normal vector n from two direction vectors d and e:

  • Use n = d × e (cross product of vectors).

• Cross product is applicable only in three-dimensional space.

Properties of the Cross Product

• The cross product is:

  • Orthogonal to both vectors involved.

  • Its magnitude equals the area of the parallelogram formed by those two vectors.

• The operation is non-commutative:

  • x × y = - (y × x).

Right Hand Rule

• For determining the direction of the cross product:

  • Point your index finger in the direction of x and your middle finger in the direction of y; your thumb will indicate the direction of x × y.

Methods for Calculating Cross Product

1. Column Vector Method:

   • Arrange vectors in column form to derive components of the cross product using determinants.

2. Grid Method:

   • Use a determinant format where basis vectors (i, j, k) are used in the top row followed by row vectors x and y below.

Results of the Cross Product

• If two vectors are parallel, then x × y = Zero vector (no area).

Vector Equations

• To move between vector point normal form and vector parametric form, recognize that:

  • d and e can be any vectors orthogonal to n.

  • Vector parametric equation format: r = r₀ + s * d + t * e.

• Direction vectors must be selected parallel to the plane (orthogonal to the normal vector n).

Area and Volume Calculations Using Cross Product

• Area of parallelogram = Length of x × y.

• Area of triangle = 0.5 * Length of x × y.