Vectors (Part 1)

Vectors

7.1 Operations on vectors (+, -, $\times$)

1. Write down, in the form $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$, the vector represented by $\overrightarrow{OP}$ if $P$ is a point of with coordinates

a) $(3,6,4)$

b) $(1,-2,-7)$

c) $(1,0,-3)$

2. $\overrightarrow{OP}$ represents a vertor r. Write down the coordinates of $P$ if

a) r = $5\mathbf{i}-7\mathbf{j}+2\mathbf{k}$

b) r = $\mathbf{i}+4\mathbf{j}$

c) r = $\mathbf{j}-\mathbf{k}$

3. If the point $A(2,-1,3)$ and $\overrightarrow{AB} = 5\mathbf{i}-7\mathbf{j}+2\mathbf{k}$, find $\overrightarrow{OB}$.

4. If a = $\mathbf{i}+\mathbf{j}+\mathbf{k}$, b = $2\mathbf{i}-\mathbf{j}+3\mathbf{k}$, c = $-\mathbf{i}+3\mathbf{j}-\mathbf{k}$, find

a) a + b

b) a - c

c) a + b + c

d) a - 2b + 3c

5. The triangle $ABC$ has its vertices at the points $A(-1,3,0)$, $B(-3,0,7)$ and $C(-1,2,3)$, Find in the form $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ of the vector

a) $\overrightarrow{AB}$

b) $\overrightarrow{AC}$

c) $\overrightarrow{CB}$

6. $A,B,C$ and $D$ are the points $(0,0,2),(-1,3,2),(1,0,4)$ and $(-1,2,-2)$ respectively. Find the vectors of

a) $\overrightarrow{AB}$

b) $\overrightarrow{BD}$

c) $\overrightarrow{CD}$

d) $\overrightarrow{AD}$

7. Given that $\overrightarrow{AB} = 3\mathbf{i}+5\mathbf{j}-4\mathbf{k}$ and $\overrightarrow{BC} = -\mathbf{i}+4\mathbf{j}-\mathbf{k}$, find $\overrightarrow{AC}$.

8. Given that $\overrightarrow{AB} = 2\mathbf{i}-4\mathbf{j}+5\mathbf{k}$ and $\overrightarrow{BC} = 3\mathbf{i}+6\mathbf{j}-2\mathbf{k}$, find $\overrightarrow{AC}$.

9. Given that $\overrightarrow{AB} = 5\mathbf{i}-7\mathbf{j}-2\mathbf{k}$ and $\overrightarrow{AC} = 2\mathbf{i}+3\mathbf{j}-2\mathbf{k}$, find $\overrightarrow{BC}$.

10. Given a = $2\mathbf{i}-\mathbf{j}-\mathbf{k}$ and b = $-\mathbf{i}+3\mathbf{j}+\mathbf{k}$.

a) Find a + b and a – b

b) Draw a diagram showing a + b  and another showing a – b

11. Given $\overrightarrow{AB} = \alpha \mathbf{i}+6\mathbf{j}+4\mathbf{k}$ , $\overrightarrow{BC} = 4\mathbf{i}+\beta \mathbf{j}-3\mathbf{k}$, and $\overrightarrow{AC} = -3\mathbf{i}+\theta \mathbf{k}$, find the values of the constants $\alpha$ , $\beta$ and $\theta$

7.2 Parallel vectors

1. Prove that the points $A(2,-1,3)$, $B(6,7,-1)$ and $C(-4,-13,9)$ are collinear

In questions 2 to 4, $\overrightarrow{OA} =$ a$= 4\mathbf{i}-12\mathbf{j}$, $\overrightarrow{OB} =$ b$= \mathbf{i}+6\mathbf{j}$.

2. Which of the following are parallel to a?

a) $\mathbf{i}+3\mathbf{j}$

b) $4\mathbf{i}-12\mathbf{j}$

c) $12\mathbf{i}-4\mathbf{j}$

d) $-4\mathbf{i}+12\mathbf{j}$

e) $\mathbf{i}-3\mathbf{j}$

3. Which of the following vectors are equal to b?

a) $2\mathbf{i}+12\mathbf{j}$

b) $-\mathbf{i}-6\mathbf{j}$

c) $\overrightarrow{AE}$ if $E(5,-6)$

d) $\overrightarrow{AF}$ if $F(6,0)$

4. If $\overrightarrow{OD}= \lambda \overrightarrow{OA}$, find the value of $\lambda$ for which $\overrightarrow{OD}+\overrightarrow{OB}$ is parallel to the $x$-axis.

5. Which of the following vectors are parallel to $3\mathbf{i}-\mathbf{j}-2\mathbf{k}$

a) $6\mathbf{i}-3\mathbf{j}-4\mathbf{k}$

b) $-9\mathbf{i}+3\mathbf{j}+6\mathbf{k}$

c) $-3\mathbf{i}-\mathbf{j}-2\mathbf{k}$

d) $-2(3\mathbf{i}+\mathbf{j}+2\mathbf{k})$

e) $\frac{3}{2}\mathbf{i}-\frac{1}{2}\mathbf{j}-\mathbf{k}$

f) $-\mathbf{i}+\frac{1}{3}\mathbf{j}+\frac{2}{3}\mathbf{k}$

6. The position vectors of the points $A,B$ and $C$ are $2\mathbf{i}-\mathbf{j}+\mathbf{k}$, $3\mathbf{i}+2\mathbf{j}-\mathbf{k}$ and $6\mathbf{i}+11\mathbf{j}-7\mathbf{k}$, respectively. Show that $A,B$ and $C$ are collinear.

7. Given that $\overrightarrow{PQ}= \pmatrix{5\\2\\-8}$ and $\overrightarrow{PR}= \pmatrix{-2\\5\\-6}$, find $\overrightarrow{QR}$

8. The position vectors of the points $P,Q$ and $R$ are $\pmatrix{5\\4\\1}$, $\pmatrix{7\\5\\4}$ and $\pmatrix{11\\7\\10}$ respectively,

     a) Find $\overrightarrow{PQ}$ and $\overrightarrow{QR}$

     b) deduce that $P,Q$ and $R$ are collinear and find the ratio $PQ:QR$

9. The coordinates of the points $A,B$ and $C$ are $(1,5,-6)$, $(3,-2,10)$ and $(7,4,18)$ respectively. Show that $A,B$ and $C$ are collinear.

10. Show that the points $P(5,4,-3)$ and $Q(3,8,-1)$ and $R(0,14,2)$ are collinear.

11. If the points $A(2,13,-5)$ and $B(3,\beta , -3)$ and $C(6,-7,\theta)$ are collinear, find the values of the constants $\beta$ and $\theta$

12. If $\mathbf{u} = 2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and $\mathbf{v} = -6\mathbf{i}+3\mathbf{j}+\lambda \mathbf{k}$, find the value of $\lambda$ when $\mathbf{u}$ and $\mathbf{v}$ are parallel.

7.3.1 Magnitude of vector

Find the magnitude of each of these vectors

1. $4\mathbf{i}+3\mathbf{j}$

2. $5\mathbf{i}-7\mathbf{j}$

3. $2\mathbf{i}-2\mathbf{j}+\mathbf{k}$

4. $6\mathbf{i}-3\mathbf{j}+4\mathbf{k}$

5. $\pmatrix{12\\5}$

6. $\pmatrix{2\\-4}$

7. $\pmatrix{-9\\7}$

8. $\pmatrix{5\\-7\\3}$

9. Find the length of the line $OP$ if $P$ is the point

     a) $(2,-1,4)$

     b) $(3,0,4)$

     c) $(-2,-2,1)$

10. Find the modulus of vector V if

     a) V = $2\mathbf{i}-4\mathbf{j}+4\mathbf{k}$

     b) V = $6\mathbf{i}+2\mathbf{j}-3\mathbf{k}$

     c) V = $11\mathbf{i}-7\mathbf{j}-6\mathbf{k}$

11. Given that v = $k\mathbf{i}+5\mathbf{j}-\sqrt{7}\mathbf{k}$ and |v|=9, find the possible value for $k$.

12. Given that $|2\mathbf{i}+k\mathbf{j}-4\mathbf{k}|=6$, find the possible value for $k$.

13. If $|k\mathbf{i}+4\mathbf{j}+4\mathbf{k}|=13$, find the possible values of $k$.

14. $A,B,C,D$ are the points with the vector $\mathbf{i}+\mathbf{j}-\mathbf{k}$, $\mathbf{i}-\mathbf{j}+2\mathbf{k}$, $\mathbf{j}+\mathbf{k}$, $2\mathbf{i}+\mathbf{j}$ respectively, find |AB| and |BD|

7.3.2 Unit vectors

Find a unit vector in the direction of each of the following vectors

1. $8\mathbf{i}-6\mathbf{j}$

2. $5\mathbf{i}-8\mathbf{j}$

3. $2\mathbf{i}+2\mathbf{j}-\mathbf{k}$

4. $6\mathbf{i}-2\mathbf{j}-3\mathbf{k}$

5. $3\mathbf{i}+4\mathbf{j}$

6. $\mathbf{i}+\mathbf{j}+4\mathbf{k}$

7. $\pmatrix{12\\5}$

8. $\pmatrix{-7\\9}$

9. $\pmatrix{5\\-7\\3}$

10. $\pmatrix{-3\\12\\-4}$

Find the coordinates of $Q$ if |OQ|=1 and $OQ$ is in the direction of:

11. $\mathbf{i}+2\mathbf{j}-2\mathbf{k}$

12. $3\mathbf{i}+2\mathbf{j}+6\mathbf{k}$

13. $8\mathbf{i}-\mathbf{j}-4\mathbf{k}$

14 $\mathbf{i}-\mathbf{j}-\mathbf{k}$

15. Find the vector v if

     a) v = $\overrightarrow{OP}$ where $P$ is the point $(0,4,5)$

     b) |v|=24 units and $\hat{\text{V}}=\frac{2}{3}\mathbf{i}+\frac{2}{3}\mathbf{j}-\frac{1}{3}\mathbf{k}$

     c) v is parallel to the vector $8\mathbf{i}+\mathbf{j}+4\mathbf{k}$ and equal in magnitude to the vector $\mathbf{i}-2\mathbf{j}+2\mathbf{k}$

16. Find $\hat{\text{r}}$ in the form $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$

     a) r = $\mathbf{i}-\mathbf{j}+\mathbf{k}$

     b) r = $5\mathbf{j}-12\mathbf{k}$

     c) r = $\mathbf{i}$

17. Given $\overrightarrow{OA}=2\mathbf{i}+3\mathbf{j}-6\mathbf{k}$ and $\overrightarrow{OC}=-2\mathbf{i}+5\mathbf{j}-2\mathbf{k}$. Find the vector which is in the same direction as $\overrightarrow{AC}$ and has magnitude $12$.

7.4 Scalar product and angles between 2 vectors

1. State whether the angle between following pairs of vectors is acute, obtuse or right angle.

     a) $(\mathbf{i}+2\mathbf{j}),(\mathbf{i}-2\mathbf{j}+5\mathbf{k})$

     b) $(\mathbf{k}),(\mathbf{i}-2\mathbf{j}-5\mathbf{k})$

     c) $\pmatrix{2\\-3\\7},\pmatrix{-1\\4\\2}$

     d) $\pmatrix{0\\-3\\1},\pmatrix{1\\1\\2}$

     e) $\pmatrix{2\\-1\\5},\pmatrix{-1\\-7\\1}$

2. Find the angle between the vectors

     a) $\pmatrix{2\\-1\\3},\pmatrix{1\\0\\-2}$

     b) $\pmatrix{-1\\2\\2},\pmatrix{2\\-3\\6}$

     c) $\pmatrix{2\\3\\1},\pmatrix{4\\-2\\-2}$

     d) $(2\mathbf{i}-7\mathbf{j}+\mathbf{k}),(\mathbf{i}+\mathbf{j}-\mathbf{k})$

3. Find the value of $a$ if the vectors $2\mathbf{i}+a\mathbf{j}-3\mathbf{k}$ and $\mathbf{i}-2\mathbf{j}+4\mathbf{k}$ are perpendicular.

4. Two vectors a and b are such that |a|=2 and |b|=4 and the angle between a and b is $\frac{1}{3}\pi$, find the angle between

     a) a and a – b

     b) b and a + b

     c) 3a – b and b

5. Given that u = $-2\mathbf{j}+2\mathbf{k}$ and v = $-3\mathbf{k}$

     a) Find |u| and |v|

     b) Find u$\cdot$v by using the definition u$\cdot$v = |u| |v| $\cos\theta$

     c) Find u$\cdot$v by using the components of the vectors

6. If r = $\pmatrix{2\\-3\\4}$ , s = $\pmatrix{1\\-7\\2}$ and t = $\pmatrix{5\\0\\-1}$, find

     a) s + t

     b) r$\cdot$s

     c) r$\cdot$t

     d) r(s + t)

     e) r$\cdot$s + r$\cdot$t

7. If u = $2\mathbf{i}+\mathbf{j}-\mathbf{k}$ and v = $\mathbf{i}+2\mathbf{j}+10\mathbf{k}$, find

     a) u$\cdot$u

     b) u$\cdot$v

     c) v$\cdot$u

     d) u(u + v)

8. Given that u = $2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and v = $-6\mathbf{i}+3\mathbf{j}+\lambda \mathbf{k}$, find the value $\lambda$ when

     a) u and v are parallel

     b) u and v are perpendicular

9. The cosine of the angle between two vectors a = $6\mathbf{i}+3\mathbf{j}-2\mathbf{k}$ and b = $-2\mathbf{i}+\lambda\mathbf{j}-4\mathbf{k}$ is $\frac{4}{21}$, find the positive value of $\lambda$

10. Show that $\mathbf{i}+7\mathbf{j}+3\mathbf{k}$ is perpendicular to both $\mathbf{i}-\mathbf{j}+2\mathbf{k}$ and $2\mathbf{i}+\mathbf{j}-3\mathbf{k}$