Academic Study Notes on Analytic Solid Geometry

ANALYTIC SOLID GEOMETRY

1.3.2 Section Ratio

• Definition: If A and B are two points with respective position vectors a and b with respect to a fixed point as the origin, and if P is a point dividing the segment AB in the ratio 2:1, the position vector p of point P can be expressed as:
  $ext{If } P ext{ divides } AB ext{ in the ratio } 2:1: p = \frac{2b + 1a}{2 + 1} = \frac{2b + a}{3}$

• General Case: If P divides AB in the ratio m:n, the formula adjusts to:
  $p = \frac{mb + na}{m+n}$
    - where m and n are scalars representing the respective ratios.

• Coordinates Example: For points A and B with coordinates A($x_1, y_1, z_1$) and B($x_2, y_2, z_2$):
    - The coordinates of P would be given by:
  $p = \frac{mb + na}{m+n} = \frac{m(x_2i + y_2j + z_2k) + n(x_1i + y_1j + z_1k)}{m+n}$
    - Resulting in component-wise coordinates:
  $p = \frac{mx_2 + nx_1}{m+n}i + \frac{my_2 + ny_1}{m+n}j + \frac{mz_2 + nz_1}{m+n}k$

Midpoint and Centroid Examples

• Midpoint of AB:
    - If position vectors of A and B are a and b, respectively, then the midpoint M has position vector:
  $M = \frac{a + b}{2}$

• Centroid of Triangle ABC:
    - For point C with position vector c:
    - The centroid G of triangle ABC has the position vector:
  $G = \frac{a + b + c}{3}$

1.4 Linear Independence

• Definition 1.4.1: A set of vectors a1, a2, …, an is said to be linearly independent if:
  $ext{If } \beta_1a_1 + \beta_2a_2 + … + \beta_n a_n = 0 ext{ implies that } \beta_1 = \beta_2 = … = \beta_n = 0$

• Linear Dependence: A set is linearly dependent if at least one of the scalars is non-zero.

• Key Note 1.4.1: Two vectors a and b are linearly dependent if:
    - There exists a scalar k such that b = ka (i.e., they are collinear); they are independent if they are not collinear.

• Key Note 1.4.2: If n non-null vectors are linearly dependent, it does not imply that any subset of them will also be dependent. If they are independent, any subset will also be independent.

Example 1.4.1

• Vectors: a = 2i + j + k, b = i + j + k, c = 4i + j + k
• Claim: Show that these vectors are linearly dependent but any two of them are independent:
    - Observation: c = a + 2b
    - Hence $la + 2b - c = 0$ with non-zero scalars. So they are dependent.

Example 1.4.2

• Vectors: a = i + 2j + k, b = 2i + 4j - 2k, c = -3i - 6j + 3k
• Claim: Show all these vectors are linearly dependent:
    - Observation: a + b + c = 0. Thus, $la + lb + lc = 0$ with non-zero scalars. Therefore they are dependent.

Theorem 1.4.1

• Statement: Three distinct points A, B, C are collinear if and only if there exist scalars x, y, z such that:
    - $xA + yB + zC = 0$ and $x + y + z = 0$

Proof

• Necessity: If points A, B, C are collinear, then C divides AB, leading to the existence of scalars such that equations above are satisfied.

Theorem 1.4.2

• Statement: Four points A, B, C, D are coplanar iff:
    - $xa + yb + zc + td = 0$ and $x + y + z + t = 0$

Proof

• This can be derived similarly as in Theorem 1.4.1.

1.5 Product of Vectors

• Types of Products:
    1. Scalar Product (Dot Product):
       - Given by: $a \bullet b = |a||b| ext{cos}( heta)$
    2. Vector Product (Cross Product):
       - Given by: $a imes b = |a||b| ext{sin}( heta) ext{ } ext{e}_{ ext{perpendicular}}$

Properties

1. Scalar product is commutative: $a \bullet b = b \bullet a$
2. For any vector a: $a \bullet a = |a|^2$
3. The scalar product is zero if vectors are perpendicular.
4. Vector product of parallel vectors is zero: $a imes a = 0$
5. Relative angle can be computed as $heta = ext{cos}^{-1}\frac{a \bullet b}{|a||b|}$

Illustrative Examples

Example 1.6.1

• Statement (i): $a imes(b + c) + b imes(c + a) + c imes(a + b) = 0$
• Solution: Using the distributive scalar property leads to simplifications.

Example 1.6.2

• Condition: If vectors satisfy $a + b + c = 0$. Then prove:

Volume of Tetrahedron

• The volume of a tetrahedron with edges represented by vectors a, b, and c is given by:
  $ext{Volume} = \frac{1}{6}|[a, b, c]|$ where [a, b, c] is the scalar triple product.
  

1.9 Vector Equations

• Equation forms involving vectors and scalars.
• Examples:
    1. $extbf{xa} + extbf{b} = 0$ leads to solution $x = -b/a$ if a  0.
• Solutions to vector equations must ensure scalar and vector operations are valid.

1.10 Vector Calculus

Definition 1.10.1

• A vector point function provides quantities at different points in space. The differentiation and integration of such functions can lead to useful calculations in geometry and physics.