Vectors and Three-Dimensional Geometry
Vectors and Three-Dimensional Geometry tests algebra with vectors, lines and planes, and geometric reasoning in 3D. Expect dot and cross products, angles, intersections, distances and skew lines.
Core rules
- Line: r=a+λd. Plane: r·n=d, or r=a+λu+μv. The normal n is perpendicular to the plane; d is the direction of the line.
- Dot product a·b=|a||b|cosθ gives angles and projections. Cross product a×b is perpendicular to both; |a×b| is the parallelogram area.
- Angle between line and plane uses sinθ=|d·n|/(|d||n|), not cos.
- Distance from point p to plane r·n=d is |p·n-d|/|n|.
- Two lines are skew if they are not parallel and do not intersect; solving their equations gives no common point.
Worked example
Plane x+2y+2z=6, point (1,1,1). Here n=(1,2,2), |n|=3. Substitute the point: 1+2+2-6=-1. Distance = |-1|/3 = 1/3. The absolute value is essential because the signed substitution can be negative.
Traps
Using a normal as a line direction; reversing cross-product order, which flips sign; treating skew lines as intersecting; forgetting absolute value in distance; using cos for line-plane angle; losing a parameter in plane equations.