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) $\lvert\mathbf u\cdot\mathbf v\rvert\leq\lvert\mathbf u\rvert$, with equality when $\theta=0$ or $\pi$ (b) $\mathbf x\times\mathbf y=\mathbf0$ (c) $[ABC]=\dfrac12\lvert\mathbf a\times\mathbf b\rvert$, independent of $\lambda$

Vectors - Scalar (Dot) Product

Vectors - Scalar (Dot) Product

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