Vectors and Matrices
Core concepts
Master vector components, magnitude, unit vectors, scalar and vector products, and geometric interpretations of dot and cross products. For matrices, know addition, multiplication, determinants, inverses, and solving linear systems. Understand that matrix multiplication is not commutative and that a zero determinant signals singularity.
Traps
- Assuming AB=BA; order matters in matrix products.
- Confusing the dot product (a scalar measuring projection) with the cross product (a vector perpendicular to both).
- Forgetting the sign convention in 2x2 and 3x3 determinants.
- Using an inverse when the determinant is zero.
Worked example
For a=(1,2,3) and b=(4,5,6), the cross product is (26-35, 34-16, 15-24)=(-3,6,-3). Its magnitude is sqrt(9+36+9)=3sqrt6, the area of the parallelogram spanned by a and b.
Rule of thumb
A system Ax=b has a unique solution exactly when det A is nonzero; otherwise check for inconsistency or infinitely many solutions.