Vector Algebra and Three-Dimensional Geometry
Vector Algebra and Three-Dimensional Geometry tests vector addition, dot and cross products, direction cosines, lines and planes, angles and distances.
Core rules
- Dot product a·b=|a||b|cos θ is a scalar; cross product a×b is a vector perpendicular to both, with |a×b|=|a||b|sin θ.
- a×b=-(b×a); a·(b×c) is a scalar triple product and gives volume of a parallelepiped.
- Direction cosines of a vector satisfy l^2+m^2+n^2=1; direction ratios are proportional numbers.
- A line is r=a+λb; a plane is r·n=d. Angle between line and plane uses the normal, not the direction vector alone.
Worked example
Let a=i+2j+3k and b=2i-j+k. Then a·b=1×2+2×(-1)+3×1=3. Also |a|=sqrt(14) and |b|=sqrt(6). Hence cos θ=3/(sqrt14 sqrt6)=3/sqrt84=sqrt21/14. The angle is acute because the dot product is positive.
Traps
- Dot product gives a scalar; writing a vector answer is wrong.
- Cross product is anticommutative, so order changes the sign.
- Direction ratios are not the same as direction cosines; normalise when cosines are required.
- Distance from a point to a plane uses the modulus in the numerator.
- Check whether the question asks for angle between lines, between line and plane, or between planes; each uses a different normal or direction.