Vectors

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are defined by
$\displaystyle \mathbf{u}=\begin{pmatrix}-1\\4\\-3\end{pmatrix}$ and $\displaystyle \mathbf{v}=\begin{pmatrix}-7\\8\\5\end{pmatrix}$
(a)  Find $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
(b)  Calculate the acute angle between $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}.$

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ have components
$\displaystyle \begin{pmatrix}p\\-2\\4\end{pmatrix}$ and $\displaystyle \begin{pmatrix}2p+16\\-3\\6\end{pmatrix}$
$\displaystyle p\in\mathbb{R}.$

(a)  (i) Find an expression for $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
      (ii) Determine the values of $\displaystyle p$ for which $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are perpendicular.
(b)  Determine the value of $\displaystyle p$ for which $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are parallel.

The vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are such that $\displaystyle |\mathbf{u}|=4$, $\displaystyle |\mathbf{v}|=5$ and $\displaystyle \mathbf{u}\cdot(\mathbf{u}+\mathbf{v})=21.$
Determine the size of the angle between the vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}.$

P and Q have coordinates (-6, 1, 2) and (-1, 11, -8) respectively.
Find the coordinates of the point R which divides PQ in the ratio 2:3.

The coordinates of points D, E and F are given by $\displaystyle D(2,-3,4)$, $\displaystyle E(1,1,-2)$ and $\displaystyle F(3,2,1).$
(a)  Express $\displaystyle \overrightarrow{ED}$ and $\displaystyle \overrightarrow{EF}$ in component form.
(b)  (i) Calculate $\displaystyle \overrightarrow{ED}.\overrightarrow{EF}.$
      (ii) Hence, or otherwise, calculate the size of angle DEF.

The vectors u and v are such that:
$\displaystyle u=\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}$
$\displaystyle v=\begin{pmatrix}1\\ 3\\ k\end{pmatrix}$
the angle between u and v is $\displaystyle 45^{\circ}$.
Find the value of k, where $\displaystyle k \gt 0$.

(a)  Show that the points $\displaystyle A(-3,2,-1),$ $\displaystyle B(6,-1,5)$ and $\displaystyle C(12,-3,9)$ are collinear.
(b)  State the ratio in which B divides AC.

E,ABCD is a rectangular-based pyramid as shown.
$\displaystyle \vec{AD}=6i+4j+2k$
$\displaystyle \vec{DC}=2i-4j+2k$
$\displaystyle \vec{DE}=-4i-3j+4k$

Express $\displaystyle \vec{BE}$ in terms of i, j and k.

(a)  Show that the points $\displaystyle D(-1,6,1)$, $\displaystyle E(1,2,7)$ and $\displaystyle F(2,0,10)$ are collinear.
Point G is such that F divides DG in the ratio 3:4.
(b)  Find the coordinates of G.

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are defined as
$\displaystyle \mathbf{u}=\begin{pmatrix}3\\ 2\\ 6\end{pmatrix}$ and $\displaystyle \mathbf{v}=\begin{pmatrix}-5\\ -2\\ 4\end{pmatrix}.$
(a)  Find $\displaystyle \mathbf{u}\cdot\mathbf{v}.$
(b)  Calculate the acute angle between $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}.$