Vectors in the plane and in space
Vector questions test both calculation and interpretation: dot products encode angles and projections, while vector equations describe lines and planes in the plane or in space. Expect intersections, perpendicularity, distance, and configurations of points.
Core rules
- Dot product: a·b=|a||b|cosθ. Thus a·b=0 iff nonzero vectors are perpendicular; the scalar projection of a on b is (a·b)/|b|.
- Line through A with direction d: r=A+td. Plane through A with normal n: (r-A)·n=0, equivalently r·n=A·n.
- A normal to a plane can be found from the cross product of two independent vectors in the plane, or by solving for coefficients from three points.
- Distance from point P to plane r·n=d is |P·n-d|/|n|.
- In space, lines may be parallel, intersecting, or skew; skew lines are not coplanar and have no intersection.
Worked example
Find the plane through A(1,0,0), B(0,1,0), C(0,0,1). Vectors AB=(-1,1,0) and AC=(-1,0,1) lie in the plane. Their cross product is (1,1,1), so take normal n=(1,1,1). Using A, the plane is x+y+z=1. The distance from the origin to this plane is |0+0+0-1|/√(1^2+1^2+1^2)=1/√3. The result is consistent with symmetry among the three intercepts.
Traps
Do not treat a dot product as a vector; it is a scalar. The normal to a plane is perpendicular to the plane, not a direction lying in it. Reverse order in a cross product changes the sign. When finding a line-plane intersection, substitute the line into the plane equation rather than equating direction vectors. Check whether a parameter solution is valid for all coordinates.