Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Vectors, Exercise 10: END-OF-CHAPTER REVIEW EXERCISE 9

With respect to the origin O, the position vectors of the points A and B are given by
$\overrightarrow{OA}=\left(\begin{array}{r}-2\\ 0\\ 6\end{array}\right) \mathrm{and} \overrightarrow{OB}=\left(\begin{array}{r}1\\ -1\\ 4\end{array}\right)$

Find a vector equation of the line $AB$.

The position vectors of $A,B\text{ and }C$ relative to an origin $O$ are given by $\overrightarrow{OA}=\left(\begin{array}{l}6\\ 3\\ 2\end{array}\right),\overrightarrow{OB}=\left(\begin{array}{r}2\\ n\\ -1\end{array}\right) \mathrm{and} \overrightarrow{OC}=\left(\begin{array}{l}8\\ 9\\ 0\end{array}\right)$, where $n$ is a constant. Find the value of $n$ for which $\left|\overrightarrow{AB}\right|=\left|\overrightarrow{CB}\right|.$

Given the vectors $8\hat{i}-2\hat{j}+5\hat{k}\text{ and } \hat{i}+2\hat{j}+p\hat{k}$ are perpendicular, find the value of the constant $p$.

The origin O and the points A, B and C are such that OABC is a rectangle. With respect to O, the position vectors of the points A and B are $-4\mathbf{i}+p\mathbf{j}-6\mathbf{k}$ and $-10\mathbf{i}-2\mathbf{j}-10\mathbf{k}\mathbf{.}$
Find the value of the positive constant p.

The points P and Q have coordinates $(0,19,-1)$ and $(-6,26,-11),$ respectively. The line L has vector equation $\mathbf{r}=3\mathbf{i}+9\mathbf{j}+2\mathbf{k}+\lambda (3\mathbf{i}-10\mathbf{j}+3\mathbf{k})$. If the perpendicular distance from Q to the line L is $\sqrt{n}$, then value of $n$ is

The points $A$ and $B$ have coordinates $(7,1,6)$ and $(10,5,1),$ respectively. The point $Q$ has coordinates $(0,-5,7)$. Find the shortest distance[in units] from $Q$ to the line $AB$.

The line $L_1$ has vector equation
$\mathbf{r}=3\mathbf{i}+2\mathbf{j}+5\mathbf{k}+\lambda (4\mathbf{i}+2\mathbf{j}+3\mathbf{k})$
The points $A(3, p, 5)$ and $B(q, 0,2),$ where $p$ and $q$ are constants, lie on the line $L_1.$
Show that $L_1$ and $L_2$ intersect and find the position vector of the point of intersection.

The line $L_1$ has vector equation $\mathbf{r}=3\mathbf{i}+2\mathbf{j}+5\mathbf{k}+\lambda (4\mathbf{i}+2\mathbf{j}+3\mathbf{k})$. The points $A(3, p, 5)$ and $B(q, 0,2),$ where $p$ and $q$ are constants, lie on the line $L_1.$ The line $L_2$ has vector equation $r=3j+k+\mu (7i+j+7k)$. Find the acute angle between $L_1$ and $L_2.$