3D vectors: lines and planes

Vector playground

<ol class="abc-list"><li><div class="list-item-paragraph">Let&nbsp;$\mathbf{u}$ and $\mathbf{v}$ be non-zero vectors such that&nbsp;$\mathbf{v}$ is a unit vector. Show that&nbsp;$|\mathbf{u}\cdot\mathbf{v}|\le|\mathbf{u}|$. When does equality occur? Explain your reasoning.<br><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Show that if&nbsp;$\mathbf{x}$ and $\mathbf{y}$ are non-zero parallel vectors, the $\mathbf{x}\times\mathbf{y}=\mathbf{0}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.<br>Let&nbsp;$C$ be a variable point with position vector $\lambda(\mathbf{a}-\mathbf{b})$.<br>Show that the area of $\triangle ABC$ is independent of $\lambda$.<span class="inline-marks">[2]</span></div></li></ol>

Let $\mathbf{u}$ and $\mathbf{v}$ be non-zero vectors such that $\mathbf{v}$ is a unit vector. Show that $|\mathbf{u}\cdot\mathbf{v}|\le|\mathbf{u}|$. When does equality occur? Explain your reasoning.


Show that if $\mathbf{x}$ and $\mathbf{y}$ are non-zero parallel vectors, the $\mathbf{x}\times\mathbf{y}=\mathbf{0}$.

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 $\lambda(\mathbf{a}-\mathbf{b})$.
Show that the area of $\triangle ABC$ is independent of $\lambda$.

Practice 1 – Basic Results Q11

Practice 1 – Basic Results Q11

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