G12 Chapter 4: Vector Algebra (Continue)

Example 8

Find the angle between the two vectors $ \begin{pmatrix} 3\\ 4 \end{pmatrix} $ and $ \begin{pmatrix} 5\\ -12 \end{pmatrix}. $

Click to view Text Book Solution
Click to view detailed MM Solution

Example 9

Given points $P(1,0,-1)$, $Q(2,4,1)$ and $R(3,5,6)$, find $\angle QPR$.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 10

Given that $\vec{a}$ and $\vec{b}$ are perpendicular vectors such that $|\vec{a}|=3$ and $|\vec{b}|=1$, evaluate $(\vec{a}-\vec{b})\cdot(\vec{a}+5\vec{b})$.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 11

Points $A$, $B$ and $C$ have position vectors

$ \vec{a}=k \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}, \qquad \vec{b}= \begin{pmatrix} 3\\ 2\\ -2 \end{pmatrix}, \qquad \vec{c}= \begin{pmatrix} 1\\ 1\\ 4 \end{pmatrix}. $

(a) Find $\overrightarrow{BC}$.

(b) Find $\overrightarrow{AB}$ in terms of $k$.

(c) Find the value of $k$ for which $\overrightarrow{AB}$ is perpendicular to $\overrightarrow{BC}$.

Click to view Text Book Solution
Click to view detailed MM Solution

Exercise 4.2

Question 1.

For $\overrightarrow{p} = \begin{pmatrix} 3\\ 2\end{pmatrix}$, $\overrightarrow{q} = \begin{pmatrix} - 1\\ 5\end{pmatrix}$ and $\overrightarrow{r} = \begin{pmatrix} - 2\\ 4\end{pmatrix}$,find:
(a) $\overrightarrow{q} \cdot \overrightarrow{p}$

Short Answer\[7\]

(b) $\overrightarrow{q}$ $\cdot$ $\overrightarrow{r}$

Short Answer\[22\]

(c) $\begin{array}{c} \overrightarrow{q}\cdot(\overrightarrow{p}+\overrightarrow{r})\end{array}$

Short Answer\[29\]

(d) $\hat{i} \cdot \overrightarrow{p}$

Short Answer\[3\]

(e) $\overrightarrow{q} \cdot \hat{j}$

Short Answer\[5\]

(f) $\hat{i} \cdot \hat{i} .$

Short Answer\[1\]

Question 2.

For $\vec{a}=\begin{pmatrix}2\\1\\3\end{pmatrix}, \vec{b}=\begin{pmatrix}-1\\1\\1\end{pmatrix}$ and $\vec{c}=\begin{pmatrix}0\\-1\\1\end{pmatrix}$, find:\\
(a) $\vec{a}\cdot\vec{b}$ \quad

Short Answer\[ 2 \]

(b) $\vec{b}\cdot\vec{a}$ \quad

Short Answer\[ 2 \]

(c) $|\vec{a}|^{2}$ \quad

Short Answer\[ 14\]

(d) $\vec{a}\cdot\vec{a}$ \quad

Short Answer\[ 14 \]

(e) $\vec{a}\cdot(\vec{b}+\vec{c})$ \quad

Short Answer\[ 4 \]

(f) $\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$

Short Answer\[ 4 \]

Question 3.

Find the angle between $\vec{m}$ and $\vec{n}$ if:
(a) $\vec{m}=\begin{pmatrix}2\\-1\\-1\end{pmatrix}$ and $\vec{n}=\begin{pmatrix}-1\\3\\2\end{pmatrix}$ \quad

Short Answer\[ 139.9^\circ \]

(b) $\vec{m}=2\hat{j}-\hat{k}$ and $\vec{n}=\hat{i}+2\hat{k}.$

Short Answer\[ 113.6^\circ \]

Question 4.

Find $t$ if the given pair of vectors are:
(i) perpendicular
(ii) parall
(a) $\overrightarrow{p} = \begin{pmatrix} 3\\ t\end{pmatrix}$and $\overrightarrow{q} = \begin{pmatrix} - 2\\ 1\end{pmatrix}$,

Short Answer\[ 6,-1.5 \]

b) $\overrightarrow{r} = \begin{pmatrix} t\\ t+ 2\end{pmatrix}$and $\overrightarrow{s} = \begin{pmatrix} t\\ - 4\end{pmatrix}$,

Short Answer\[ t=2 \pm 2\sqrt{3},-6 \]

(c) $\overrightarrow{a} = \begin{pmatrix} 0\\ t+ 2\end{pmatrix}$and$\overrightarrow{b}=\begin{pmatrix}2-3t\\t\end{pmatrix}$

Short Answer\[ 0,-2;2/3 \]

Question 5.

Find $t$ if $\begin{pmatrix}3\\t\\-2\end{pmatrix}$ is perpendicular to $\begin{pmatrix}1-t\\-3\\4\end{pmatrix}.$

Short Answer\[ t = \frac{-5}{6} \]

Question 6.

$ABCD$ is a parallelogram with $AB$ parallel to $DC.$ Let $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{AD}=\overrightarrow{b}.$
(a) Express $\overrightarrow{AC}$ and $\overrightarrow{BD}$ in terms of $\overrightarrow{a}$ and $\overrightarrow{b}$

Short Answer\[ \vec{b} - \vec{a} \]

(b) Simplify $( \overrightarrow{a} + \overrightarrow{b} ) ^{\cdot \cdot }( \overrightarrow{b} - \overrightarrow{a} )$

Short Answer\[|\vec{b}|^2 - |\vec{a}|^2 \]

(c) Hence show that if $ABCD$ is a rhombus then its diagonals are perpendicular.

Short AnswerProved

Example 12

Find the area of the parallelogram determined by the vectors $ \vec{a}= \begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} $ and $ \vec{b}= \begin{pmatrix} 1\\ 4\\ -1 \end{pmatrix}. $

Click to view Text Book Solution
Click to view detailed MM Solution

Example 13

Find the area of the triangle $ABC$ with vertices $A(1,-1,3)$, $B(0,4,1)$ and $C(2,7,2)$.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 14

(a) Calculate $\vec{a}\times\vec{b}$ when $\vec{a}=3\hat{i}+2\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-4\hat{j}+2\hat{k}$.

(b) Find a unit vector $\hat{n}$ that is perpendicular to both $\vec{a}$ and $\vec{b}$.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 15

Given that $|\vec{a}|=4$, $|\vec{b}|=5$ and that $\vec{a}$ and $\vec{b}$ are perpendicular, evaluate $ |(2\vec{a}-\vec{b})\times(\vec{a}+3\vec{b})|. $

Click to view Text Book Solution
Click to view detailed MM Solution

Exercise 4.3

Question 1.

Find a vector perpendicular to the following pair of vectors:
(a) $\begin{pmatrix}3\\1\\1\end{pmatrix}$ and $\begin{pmatrix}1\\2\\3\end{pmatrix}$ \quad

Short Answer\[ \begin{pmatrix}1\\-8\\5\end{pmatrix} \]

(b) $\begin{pmatrix}3\\-1\\4\end{pmatrix}$ and $\begin{pmatrix}-1\\1\\5\end{pmatrix}$

Short Answer\[ \begin{pmatrix}-9\\-19\\2\end{pmatrix} \]

Question 2.

Consider$\begin{array}{c}{\overrightarrow{a}}=\left(\begin{array}{c}{2}\\{-1}\\{3}\end{array}\right)\:\mathrm{and}\:\overrightarrow{b}=\left(\begin{array}{c}{1}\\{0}\\{-1}\end{array}\right).\end{array}$
(a) Find$\overrightarrow a\times\overrightarrow b.$

Short Answer\[ \begin{pmatrix}1\\5\\1\end{pmatrix} \]

(b) Find sin$\theta$ using$\mid\overrightarrow a\times\overrightarrow b\mid=\mid\overrightarrow a\mid\mid\overrightarrow b\mid\sin\theta.$

Short Answer\[ \frac{3\sqrt{21}}{14} \]

Question 3.

Prove that for any two vectors $\overrightarrow{a}$ and $\overrightarrow{b} $, $| \overrightarrow{a} \times \overrightarrow{b} | ^2+ ( \overrightarrow{a} \cdot \overrightarrow{b} ) ^2= | \overrightarrow{a} | ^2$ $| \overrightarrow{b} | ^2.$

Short Answer\[ \]

Question 4.

Given points $A,B$ and $C$ with coordinates (3,-5,1),(7,7,2) and (-1,1,3).
(a)Calculate$\overrightarrow p=\overrightarrow AB\times\overrightarrow AC$ and$\overrightarrow q=\overrightarrow BA\times\overrightarrow BC.$

Short Answer\[ \begin{pmatrix}-18\\12\\-72\end{pmatrix} \]

(b) What can you say about vectors $\overrightarrow p$ and $\overrightarrow q?$

Short Answer\[ \vec p=-\vec q \]

Question 5.

The points $A(3,1,2),B(-1,1,5)$ and $C(7,2,3)$ are vertices of a parallelogram $ABCD.$
(a) Find the coordinates of $D.$

Short Answer\[ D(11,2,0) \]

(b) Calculate the area of the parallelogram.

Short Answer\[ \sqrt{281} \]

Example 16

Find the Cartesian equation of the line with vector equation $ \vec{r} = \begin{pmatrix} 1\\ 4\\ -1 \end{pmatrix} + t \begin{pmatrix} 3\\ 2\\ 5 \end{pmatrix}. $

Click to view Text Book Solution
Click to view detailed MM Solution

Example 17

Does the point $A(3,-2,2)$ lie on the line with equation $ \frac{x+1}{2} = \frac{4-y}{3} = \frac{2z}{3} \;? $

Click to view Text Book Solution
Click to view detailed MM Solution

Example 18

Find the vector equation of the plane containing points $M(2,2,-2)$, $N(1,-1,3)$ and $P(4,0,2)$.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 19

Find the Cartesian equation of the plane containing points $M(2,2,-2)$, $N(1,-1,3)$ and $P(4,0,2)$.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 20

Determine whether points $A(3,-1,4)$, $B(2,1,1)$, $C(4,3,1)$ and $D(-3,1,4)$ lie in the same plane.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 21

Find a vector equation of the plane containing the line $ \vec{r} = \begin{pmatrix} -2\\ 1\\ 2 \end{pmatrix} + t \begin{pmatrix} -1\\ 1\\ 1 \end{pmatrix} $ and point $A(3,-1,2)$.

Click to view Text Book Solution
Click to view detailed MM Solution

Example 22

Vector $ \vec{n} = \begin{pmatrix} 2\\ 4\\ -2 \end{pmatrix} $ is perpendicular to the plane which contains point $A(1,-5,2)$.

(a) Write an equation of the plane in the form $\vec{r}\cdot\vec{n}=d$.

(b) Find the Cartesian equation of the plane.

Click to view Text Book Solution
Click to view detailed MM Solution

================

Exercise 4.4

Question 1.

Find the vector equation of the line:
(a) parallel to $\begin{pmatrix}2\\1\\3\end{pmatrix}$ and through the point $(1,3,-7)$

Short Answer\[ \vec{r} = \begin{pmatrix}1\\3\\-7\end{pmatrix} + t\begin{pmatrix}2\\1\\3\end{pmatrix} \]

(b) through $(0,1,2)$ and with direction vector $\hat{i}+\hat{j}-2\hat{k}.$

Short Answer\[ \vec{r} = \begin{pmatrix}0\\1\\2\end{pmatrix} + t\begin{pmatrix}1\\1\\-2\end{pmatrix} \]

(c) parallel to the $x$-axis and through the point $(-2,2,2).$

Short Answer\[ \vec{r} = \begin{pmatrix}-2\\2\\2\end{pmatrix} + t\begin{pmatrix}1\\0\\0\end{pmatrix} \]

Question 2.

(a) Find the Cartesian equation of the line with parametric equation $x= 3t+ 1$, $y= 4- 2t$, $z= 3t- 1.$

Short Answer\[ \frac{x-1}{3} = \frac{y-4}{-2} = \frac{z+1}{3} \]

(b) Find the unit vector in the direction of the line

Short Answer\[ \hat{d} = \frac{1}{\sqrt{22}}\begin{pmatrix}3\\-2\\3\end{pmatrix} \]

Question 3.

Find the equation of the plane:
(a) with normal vector$\begin{pmatrix}2\\-1\\3\end{pmatrix}$ and which passes through $(-1,2,4)$

Short Answer\[ 2x - y + 3z - 8 = 0 \]

(b) perpendicular to the line joining points $A(2,3,1)$ and $B(5,7,2)$ and which passes through $A.$

Short Answer\[ 3x + 4y + z - 19 = 0 \]

(c) containing $A(3,2,1)$ and the line $x=1+t,y=2-t,z=3+2t.$

Short Answer\[ x + 3y + z - 10 = 0 \]

Question 4.

Find the equation of the plane through $A(-1,2,1),B(4,1,1)$ and $C(2,0,3):$
(a) in vcctor form

Short Answer\[ \vec{r} = \begin{pmatrix}-1\\2\\1\end{pmatrix} + t_1\begin{pmatrix}5\\-1\\0\end{pmatrix} + t_2\begin{pmatrix}3\\-2\\2\end{pmatrix} \]

(b) in Cartesian form.

Short Answer\[ 2x + 10y + 7z - 25 = 0 \]

Question 5.

Find the Cartcsian equation of the plane with vector equation $$\overrightarrow{r}=\begin{pmatrix}1\\2\\3\end{pmatrix}+t_1\begin{pmatrix}1\\1\\2\end{pmatrix}+t_2\begin{pmatrix}2\\-1\\5\end{pmatrix}.$$

Short Answer\[ 7x - y - 3z + 4 = 0 \]