Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 11 / a

This is a plane through the origin in three-dimensional Euclidean space, with nonzero normal vector $(1,2,-3{)}^T$:

For example, chooseBoth vectors satisfy the plane equation. They are linearly independent vectors: in $\alpha \mathbf{u}+\beta \mathbf{v}=0$, the second coordinate gives $\alpha =0$ and the third gives $\beta =0$.

No. Every vector in the plane can be written asThe two linearly independent vectors in part (ii) therefore form a vector space basis for this vector subspace. Its dimension is $2$, so any three vectors in it have linear dependence.