Let $\mathbf{u}$ and $\mathbf{v}$ be non-zero vectors such that $\mathbf{v}$ is a un

Question

Let $\mathbf{u}$ and $\mathbf{v}$ be non-zero vectors such that $\mathbf{v}$ is a unit vector. Show that $\left| \mathbf{u}\cdot \mathbf{v} \right|\le \left| \mathbf{u} \right|$. When does equality occur? Explain your reasoning.[3] Show that if $\mathbf{x}$ and $\mathbf{y}$ are non-zero parallel vectors, then $\mathbf{x}\times \mathbf{y}=\mathbf{0}$.[2] Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Let $C$ be a variable point with position vector $\lambda \left( \mathbf{a}-\mathbf{b} \right)$. Show that the area of $\vartriangle ABC$ is independent of $\lambda $.[2]

Accepted Answer

(a) |u · v| ≤ |u|, equality when u ∥ v; (b) x × y = 0 (shown); (c) Area = ½|a × b|, independent of λ

Vectors - Vectors - General

Vectors - Vectors - General

Video Solution: