feat(sql): Add comprehensive vector mathematics support

Implement native vector data type and complete vector math function library for
physics calculations, geometry, line analysis, and scientific computations.

New Vector DataType:
- Added Vector(Vec<f64>) variant to DataValue enum
- Display format: "[x,y,z]"
- Full integration with JSON, comparisons, serialization, and aggregates
- Parse vectors from "[1,2,3]" or "1 2 3" format

Vector Functions (10 total):
- VEC(x, y, z, ...) - Construct vectors from numeric components
- VEC_ADD(v1, v2) - Element-wise vector addition
- VEC_SUB(v1, v2) - Element-wise vector subtraction
- VEC_SCALE(v, scalar) - Scalar multiplication
- VEC_DOT(v1, v2) - Dot product (returns Float)
- VEC_MAG(v) - Magnitude/length of vector
- VEC_NORMALIZE(v) - Normalize to unit length
- VEC_DISTANCE(v1, v2) - Euclidean distance between vectors
- VEC_CROSS(v1, v2) - Cross product (3D only)
- VEC_ANGLE(v1, v2) - Angle between vectors in radians

Real-World Applications:
- Physics: velocity, displacement, force calculations
- Geometry: line intersection, parallel/perpendicular detection
- Point-to-line distance calculations
- Angle measurements and direction analysis
- Collinearity testing
- 3D graphics and transformations

Examples added:
- examples/vector_math_demo.sql - Basic vector operations
- examples/line_intersection_demo.sql - Line geometry analysis
- examples/line_relationships_summary.sql - Comprehensive line relationships

Usage:
  SELECT VEC(1, 2, 3) as position;
  SELECT VEC_ADD(VEC(10,5,0), VEC(5,5,0)) as displacement;
  SELECT VEC_DOT(VEC(1,2,3), VEC(4,5,6)) as dot_product;
  SELECT VEC_MAGNITUDE(VEC(3,4)) as distance;
  SELECT VEC_CROSS(VEC(1,0,0), VEC(0,1,0)) as perpendicular;

All vector functions accessible via --list-functions and --function-help.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

Author: TimelordUK

examples/line_intersection_demo.sql

@@ -0,0 +1,133 @@
+-- Line-Line Intersection Demonstration
+-- Shows how to determine if two lines intersect using vector math
+
+-- Example 1: Two intersecting lines in 2D
+-- Line 1: from (0,0) to (4,4)
+-- Line 2: from (0,4) to (4,0)
+SELECT
+    VEC(0,0) as line1_p1,
+    VEC(4,4) as line1_p2,
+    VEC(0,4) as line2_p1,
+    VEC(4,0) as line2_p2,
+
+    -- Direction vectors
+    VEC_SUB(VEC(4,4), VEC(0,0)) as line1_dir,
+    VEC_SUB(VEC(4,0), VEC(0,4)) as line2_dir,
+
+    -- For 2D lines, use cross product with z=0 to check if parallel
+    -- If cross product is 0, lines are parallel
+    VEC_CROSS(
+        VEC_SUB(VEC(4,4,0), VEC(0,0,0)),
+        VEC_SUB(VEC(4,0,0), VEC(0,4,0))
+    ) as cross_product,
+
+    -- Magnitude of cross product (0 = parallel, non-zero = intersecting/skew)
+    VEC_MAG(VEC_CROSS(
+        VEC_SUB(VEC(4,4,0), VEC(0,0,0)),
+        VEC_SUB(VEC(4,0,0), VEC(0,4,0))
+    )) as parallel_check;
+GO
+
+-- Example 2: Parallel lines (should have cross product magnitude of 0)
+-- Line 1: from (0,0) to (4,2)
+-- Line 2: from (0,2) to (4,4) - same direction!
+SELECT
+    VEC(0,0) as line1_p1,
+    VEC(4,2) as line1_p2,
+    VEC(0,2) as line2_p1,
+    VEC(4,4) as line2_p2,
+
+    VEC_SUB(VEC(4,2), VEC(0,0)) as line1_dir,
+    VEC_SUB(VEC(4,4), VEC(0,2)) as line2_dir,
+
+    -- Check if parallel using 2D cross product (extend to 3D with z=0)
+    VEC_CROSS(
+        VEC_SUB(VEC(4,2,0), VEC(0,0,0)),
+        VEC_SUB(VEC(4,4,0), VEC(0,2,0))
+    ) as cross_product,
+
+    VEC_MAG(VEC_CROSS(
+        VEC_SUB(VEC(4,2,0), VEC(0,0,0)),
+        VEC_SUB(VEC(4,4,0), VEC(0,2,0))
+    )) as is_parallel;
+GO
+
+-- Example 3: Perpendicular lines (dot product of directions should be 0)
+-- Line 1: horizontal (1,0)
+-- Line 2: vertical (0,1)
+SELECT
+    VEC(1,0) as line1_dir,
+    VEC(0,1) as line2_dir,
+
+    VEC_DOT(VEC(1,0), VEC(0,1)) as dot_product,
+
+    -- Angle should be 90 degrees (π/2 radians)
+    VEC_ANGLE(VEC(1,0), VEC(0,1)) as angle_rad,
+    VEC_ANGLE(VEC(1,0), VEC(0,1)) * 180 / PI() as angle_deg;
+GO
+
+-- Example 4: 3D line intersection check
+-- Two lines in 3D space
+SELECT
+    VEC(0,0,0) as line1_p1,
+    VEC(1,1,1) as line1_p2,
+    VEC(0,1,0) as line2_p1,
+    VEC(1,0,0) as line2_p2,
+
+    -- Direction vectors
+    VEC_SUB(VEC(1,1,1), VEC(0,0,0)) as line1_dir,
+    VEC_SUB(VEC(1,0,0), VEC(0,1,0)) as line2_dir,
+
+    -- Cross product (perpendicular to both)
+    VEC_CROSS(
+        VEC_SUB(VEC(1,1,1), VEC(0,0,0)),
+        VEC_SUB(VEC(1,0,0), VEC(0,1,0))
+    ) as perpendicular_vec,
+
+    -- If magnitude is 0, lines are parallel
+    VEC_MAG(VEC_CROSS(
+        VEC_SUB(VEC(1,1,1), VEC(0,0,0)),
+        VEC_SUB(VEC(1,0,0), VEC(0,1,0))
+    )) as parallel_check;
+GO
+
+-- Example 5: Angle between two line segments
+-- Useful for detecting sharp vs smooth intersections
+SELECT
+    VEC(1,0) as dir1,
+    VEC(1,1) as dir2,
+
+    -- Angle between lines
+    VEC_ANGLE(VEC(1,0), VEC(1,1)) as angle_rad,
+    VEC_ANGLE(VEC(1,0), VEC(1,1)) * 180 / PI() as angle_deg,
+
+    -- Normalize first to ensure we're comparing directions
+    VEC_ANGLE(
+        VEC_NORMALIZE(VEC(1,0)),
+        VEC_NORMALIZE(VEC(1,1))
+    ) * 180 / PI() as normalized_angle;
+GO
+
+-- Example 6: Distance from point to line
+-- Using the formula: distance = ||(p - a) × dir|| / ||dir||
+-- where p is the point, a is a point on the line, dir is line direction
+SELECT
+    VEC(2,2) as point,
+    VEC(0,0) as line_point,
+    VEC(1,0) as line_dir,
+
+    -- Vector from line point to target point
+    VEC_SUB(VEC(2,2), VEC(0,0)) as to_point,
+
+    -- Cross product (needs to be 3D)
+    VEC_CROSS(
+        VEC_SUB(VEC(2,2,0), VEC(0,0,0)),
+        VEC(1,0,0)
+    ) as cross_vec,
+
+    -- Distance = magnitude of cross / magnitude of direction
+    VEC_MAG(VEC_CROSS(
+        VEC_SUB(VEC(2,2,0), VEC(0,0,0)),
+        VEC(1,0,0)
+    )) / VEC_MAG(VEC(1,0,0)) as distance_to_line;
+GO

examples/line_relationships_summary.sql

@@ -0,0 +1,184 @@
+-- LINE RELATIONSHIPS: What We Can Determine
+-- Using vector math to analyze relationships between lines
+
+-- =============================================================================
+-- 1. PARALLEL LINES CHECK
+-- =============================================================================
+-- Two lines are parallel if their direction vectors are parallel
+-- Test: Cross product magnitude = 0
+
+SELECT '=== PARALLEL LINES ===' as test;
+GO
+
+-- Parallel lines (same direction)
+SELECT
+    'Parallel' as relationship,
+    VEC(0,0) as line1_start,
+    VEC(2,2) as line1_end,
+    VEC(1,1) as line2_start,
+    VEC(3,3) as line2_end,
+
+    VEC_MAG(VEC_CROSS(
+        VEC(2,2,0),  -- line1 direction
+        VEC(2,2,0)   -- line2 direction (same!)
+    )) as cross_mag,
+
+    CASE
+        WHEN VEC_MAG(VEC_CROSS(VEC(2,2,0), VEC(2,2,0))) < 0.0001
+        THEN 'PARALLEL'
+        ELSE 'NOT PARALLEL'
+    END as result;
+GO
+
+-- =============================================================================
+-- 2. INTERSECTING LINES CHECK
+-- =============================================================================
+-- Two lines intersect if they're NOT parallel (cross product ≠ 0)
+
+SELECT '=== INTERSECTING LINES ===' as test;
+GO
+
+-- Classic X intersection
+SELECT
+    'Intersecting' as relationship,
+    VEC(0,0) as line1_start,
+    VEC(4,4) as line1_end,
+    VEC(0,4) as line2_start,
+    VEC(4,0) as line2_end,
+
+    VEC_MAG(VEC_CROSS(
+        VEC(4,4,0),   -- line1 direction
+        VEC(4,-4,0)   -- line2 direction
+    )) as cross_mag,
+
+    CASE
+        WHEN VEC_MAG(VEC_CROSS(VEC(4,4,0), VEC(4,-4,0))) > 0.0001
+        THEN 'INTERSECT'
+        ELSE 'PARALLEL'
+    END as result,
+
+    -- Expected intersection point is (2,2) for this case
+    'Expected: (2,2)' as note;
+GO
+
+-- =============================================================================
+-- 3. PERPENDICULAR LINES CHECK
+-- =============================================================================
+-- Two lines are perpendicular if dot product of directions = 0
+
+SELECT '=== PERPENDICULAR LINES ===' as test;
+GO
+
+SELECT
+    'Perpendicular' as relationship,
+    VEC(1,0) as line1_dir,
+    VEC(0,1) as line2_dir,
+
+    VEC_DOT(VEC(1,0), VEC(0,1)) as dot_product,
+
+    CASE
+        WHEN ABS(VEC_DOT(VEC(1,0), VEC(0,1))) < 0.0001
+        THEN 'PERPENDICULAR'
+        ELSE 'NOT PERPENDICULAR'
+    END as result,
+
+    VEC_ANGLE(VEC(1,0), VEC(0,1)) * 180 / PI() as angle_degrees;
+GO
+
+-- =============================================================================
+-- 4. ANGLE BETWEEN LINES
+-- =============================================================================
+-- Can determine the angle of intersection
+
+SELECT '=== ANGLE BETWEEN LINES ===' as test;
+GO
+
+SELECT
+    'Acute angle' as case_type,
+    VEC(1,0) as line1_dir,
+    VEC(1,1) as line2_dir,
+    VEC_ANGLE(VEC(1,0), VEC(1,1)) * 180 / PI() as angle_deg;
+GO
+
+SELECT
+    'Right angle' as case_type,
+    VEC(1,0) as line1_dir,
+    VEC(0,1) as line2_dir,
+    VEC_ANGLE(VEC(1,0), VEC(0,1)) * 180 / PI() as angle_deg;
+GO
+
+SELECT
+    'Obtuse angle' as case_type,
+    VEC(1,0) as line1_dir,
+    VEC(-1,1) as line2_dir,
+    VEC_ANGLE(VEC(1,0), VEC(-1,1)) * 180 / PI() as angle_deg;
+GO
+
+-- =============================================================================
+-- 5. DISTANCE FROM POINT TO LINE
+-- =============================================================================
+-- Can compute perpendicular distance from any point to a line
+
+SELECT '=== POINT-TO-LINE DISTANCE ===' as test;
+GO
+
+SELECT
+    VEC(2,2) as point,
+    'y=0 (x-axis)' as line,
+    VEC(0,0) as line_point,
+    VEC(1,0) as line_dir,
+
+    VEC_MAG(VEC_CROSS(
+        VEC(2,2,0),
+        VEC(1,0,0)
+    )) / VEC_MAG(VEC(1,0,0)) as distance,
+
+    'Expected: 2' as verification;
+GO
+
+-- =============================================================================
+-- 6. COLLINEAR CHECK
+-- =============================================================================
+-- Three points are collinear if cross product of vectors between them = 0
+
+SELECT '=== COLLINEAR POINTS ===' as test;
+GO
+
+SELECT
+    'Collinear' as case_type,
+    VEC(0,0) as p1,
+    VEC(1,1) as p2,
+    VEC(2,2) as p3,
+
+    VEC_MAG(VEC_CROSS(
+        VEC_SUB(VEC(1,1,0), VEC(0,0,0)),
+        VEC_SUB(VEC(2,2,0), VEC(0,0,0))
+    )) as cross_mag,
+
+    CASE
+        WHEN VEC_MAG(VEC_CROSS(
+            VEC_SUB(VEC(1,1,0), VEC(0,0,0)),
+            VEC_SUB(VEC(2,2,0), VEC(0,0,0))
+        )) < 0.0001
+        THEN 'COLLINEAR'
+        ELSE 'NOT COLLINEAR'
+    END as result;
+GO
+
+-- =============================================================================
+-- SUMMARY: What we CAN determine
+-- =============================================================================
+-- ✓ Are lines parallel?
+-- ✓ Are lines perpendicular?
+-- ✓ Do lines intersect (in 2D/3D)?
+-- ✓ What angle do lines make?
+-- ✓ Distance from point to line
+-- ✓ Are three points collinear?
+-- ✓ Direction of lines
+-- ✓ Relative orientation (cross product sign)
+
+-- What would make this even more powerful:
+-- • Computing exact intersection point (needs parametric solving)
+-- • Closest point on line to another point
+-- • Line-segment intersection (requires bounds checking)
+-- • 3D skew lines distance