Vector Algebra and Three-Dimensional Geometry
Vector Algebra and Three-Dimensional Geometry
This domain tests vectors in component form, products, and spatial lines and planes. A vector has magnitude and direction; its position vector from the origin is written with i, j, k components. The dot product gives a scalar and measures projection; the cross product gives a vector perpendicular to both.
Core rules
- If a=a1i+a2j+a3k, then |a|=√(a1^2+a2^2+a3^2); unit vector is a/|a|.
- a·b=a1b1+a2b2+a3b3=|a||b|cosθ; perpendicular vectors have dot product 0.
- a×b is a vector; |a×b|=|a||b|sinθ; parallel vectors have cross product 0.
- Line: r=a+λb, where a is a point vector and b is a direction vector.
- Plane: r·n=d, where n is a normal vector.
- Direction cosines satisfy l^2+m^2+n^2=1.
Worked example
Let a=i+2j+2k and b=2i−j+2k. Then |a|=√(1+4+4)=3 and |b|=√(4+1+4)=3. The dot product is 1(2)+2(−1)+2(2)=2−2+4=4. Thus cosθ=4/(3×3)=4/9, and θ=cos^{-1}(4/9). The positive dot product means the angle between the vectors is acute.
Traps
Treating a dot product as a vector; using direction ratios as if they were already direction cosines; forgetting absolute value in a distance formula; confusing line direction with plane normal; missing that the angle between lines is usually taken as acute.