Vector Algebra and Three-Dimensional Geometry - BSEB Inter Study Guide
Vector Algebra and Three-Dimensional Geometry
Questions test dot and cross products, projections, direction cosines, lines and planes in space. The exam favours exact angle and distance calculations.
Core rules
- a·b=|a||b|cosθ; a×b=|a||b|sinθ n̂, with a×b=−b×a.
- If a·b=0, vectors are perpendicular; if a×b=0, they are parallel.
- Projection of a on b is (a·b)/|b|.
- Direction cosines of vector a are a_x/|a|, a_y/|a|, a_z/|a|, and l²+m²+n²=1.
- Line through a with direction b: r=a+λb. Plane: r·n=d.
- Angle between line and plane satisfies sinθ=|b·n|/(|b||n|).
Worked example
Find the angle between a=i+j and b=j+k. a·b=0·0+1·1+1·0=1. |a|=√2, |b|=√2. cosθ=1/(√2·√2)=1/2, so θ=π/3.
Traps
- Reversing the cross product order and losing the sign.
- Using cosθ for the angle between a line and a plane; the normal gives sinθ.
- Confusing direction ratios with direction cosines.
- Forgetting to take the modulus in projection and angle formulas.
- Writing a plane equation with a normal that is not perpendicular to the plane.