Week 11 Exercises

11.4 Sheet 21 Vectors I

Exercise 1

Two vectors \(\underline A\) and \(\underline B\) are at right angles. Find

  1. the magnitude of the vector \(\underline A+\underline B\) if \(|\underline A| = 1\) and \(|\underline B| = 2\).
  2. the magnitude of the vector \(\underline A+\underline B\) if \(|\underline A|=20\) and \(|\underline B| = 21\).

Exercise 2

Determine the resultant of the vectors \(\underline A, \underline B, \underline C\) and \(\underline D\) given that \(\underline A = 20\) km \(3o^\circ\) S of E, \(\underline B = 50\) km due W, \(\underline C = 40\) km NE and \(\underline D=30\) km SW.

Exercise 3

An object \(P\) is acted on by three forces as shown on Figure 11.1. All forces lie in the same plane (are co-planar). Determine the required force and its direction so as to prevent \(P\) from moving.
resultant force

Figure 11.1: resultant force

Exercise 4

For the triangle on Figure 11.2, prove by vectors that the line \(DE\), where \(D\) and \(E\) are the mid-points of \(AB\) and \(AC\) respectively, is half the length of the side \(BC\).
triangles

Figure 11.2: triangles

Exercise 5

The quadrilateral \(ABCD\) shown on Figure 11.3 has the mid-points of the diagonals as \(P\) and \(Q\). Show that \(\underline{AB}+\underline{AD}+\underline{CB}+\underline{CD} = 4\underline{PQ}\)
quadrilateral

Figure 11.3: quadrilateral

[Solutions: 1. (i) \(\sqrt{5}\), (ii) \(29\); 2. \([25.7 \angle -173^{\circ}]\); 3. \(-323\underline{i}\,N\);]

11.5 Sheet 22 Vectors II

Exercise 1

If \(\underline{z_1} = 5\underline{i}-2\underline{j}, \underline{z_2} = 3\underline{i}+3\underline{j}\) and \(\underline{z_3}=4\underline{i}=\underline{j}\), determine \(\underline{z_1}+\underline{z_2}+\underline{z_3}\) and \(\underline{z_1}-\underline{z_2}-\underline{z_3}\).

Exercise 2

If \(\underline{OA}=4\underline{i}+3\underline{j}\) and \(\underline{OB}=5\underline{i}-2\underline{j}\) determine \(\underline{AB}\).

Exercise 3

If \(\underline{OA} = 4\underline{i}+3\underline{j}, \underline{OB}=6\underline{i}-2\underline{j}\) an \(\underline{OC}=2\underline{i}-\underline{j}\) determine \(\underline{AB}, \underline{BC}\) and \(\underline{CA}\) and establish the lengths of the sides of the triangle \(ABC\).

Exercise 4

Determine the magnitude and direction cosines of each of the vectors \(3\underline{i}+7\underline{j}-4\underline{k}, \underline{i}-5\underline{j}-8\underline{k}\) and \(6\underline{i}-2\underline{j}+12\underline{k}\) and then determine the magnitude and direction cosines of their sum.

Exercise 5

If \(\underline a=2\underline{i}+4\underline{j}-3\underline{k}\) and \(\underline b=\underline{i}+3\underline{j}+2\underline{k}\) establish the value of the angle between \(\underline a\) and \(\underline b\).

Exercise 6

If \(\underline{OA}=2\underline{i}+3\underline{j}-\underline{k}\) and \(\underline{OB}=\underline{i}-2\underline{j}+3\underline{k}\) establish the value of the angle between \(\underline{OA}\) and \(\underline{OB}\).

Exercise 7

Given that the position vectors describing points \(P\) and \(Q\) are \(\underline{i}+3\underline{j}-7\underline{k}\) and \(5\underline{i}-2\underline{j}-6\underline{k}\) respectively, find the vector \(\underline{PQ}\) and determine its direction cosines.

Exercise 8

If \(\vec{a}=\frac2{11}\underline{i}+p\underline{j}+\frac9{11}\underline{k}\) is a unit vector, what is the value of \(p\)?

Exercise 9

Find the unit vector \(\hat{\underline{a}}\) parallel to the vector \(\underline a=3\underline{i}-2\underline{j}+\underline{k}\).

[Solutions: 1. \(12\underline{i}\), \(-2\underline{i}-4\underline{j}\); 2. \(\underline{i}-5\underline{j}\); 3. \(2\underline{i}-5\underline{j}\), \(-4\underline{i}+\underline{j}\), \(2\underline{i}+4\underline{j}\), \(\sqrt{29}\), \(\sqrt{17}\), \(\sqrt{20}\); 4. \(\sqrt{74}\), \(3\sqrt{10}\), \(2\sqrt{46}\), \(10\) are magnitudes; 5. \(66.7^{\circ}\); 6. \(120^{\circ}\); 7. \(\underline{PQ}=4\underline{i}-5\underline{j}+\underline{k}\), direction cosines are \(4/\sqrt{42}\), \(-5/\sqrt{42}\), \(1/\sqrt{42}\); 8. \(p=6/11\); 9. \(|\underline{a}|=\sqrt{14}\), so \(\hat{\underline{a}}=\frac{3}{\sqrt{14}}\underline{i} - \frac{2}{\sqrt{14}}\underline{j} + \frac{1}{\sqrt{14}}\underline{k}\);]

11.6 Sheet 23 Vectors III

Exercise 1

If \(\underline{a} = 2\underline{i} + 4\underline{j} - 3\underline{k}\) and \(\underline{b} = \underline{i} + 3\underline{j} + 2\underline{k}\), calculate the scalar and vector product and the angle between the vectors \(\underline{a}\) and \(\underline{b}\).

Exercise 2

Given that \(\underline{OA} = 2\underline{i} + 3\underline{j} - \underline{k}\) and \(\underline{OB} = \underline{i} - 2\underline{j} + 3\underline{k}\) determine

  1. the scalar product \(\underline{OA}\cdot\underline{OB}\),
  2. the vector product \(\underline{OA}\times\underline{OB}\),
  3. the angle between \(\underline{OA}\) and \(\underline{OB}\).

Exercise 3

Determine the scalar and vector product of \(\underline{a}\) and \(\underline{b}\) when

  1. \(\underline{a} = \underline{i} + 2\underline{j} - \underline{k}\) and \(\underline{b} = 2\underline{i} + 3\underline{j} + \underline{k}\),
  2. \(\underline{a} = 2\underline{i} + 3\underline{j} + 4\underline{k}\) and \(\underline{b} = 5\underline{i} - 2\underline{j} + \underline{k}\).

Exercise 4

Use the scalar product to find the angle between the vectors \(\underline{a}=2\underline{i} + 3\underline{j} - \underline{k}\) and \(\underline{b}=3\underline{i} - 5\underline{j} + 2\underline{k}\).

Exercise 5

If \(\underline{a} = 3\underline{i} - \underline{j} + 2\underline{k}\) and \(\underline{b} = \underline{i} + 3\underline{j} - 2\underline{k}\), determine the magnitude and direction cosines of the vector product \(\underline{a} \times \underline{b}\).

Exercise 6

If \(\underline{a} = 5\underline{i} + 4\underline{j} + 2\underline{k}, \underline{b} = 4\underline{i} - 5\underline{j} + 3\underline{k}\) and \(\underline{c} = 2\underline{i} - \underline{j} - 2\underline{k}\) determine

  1. the value of \(\underline{a} \cdot \underline{b}\) and the angle between the vectors \(\underline{a}\) and \(\underline{b}\),
  2. the magnitude and direction cosines of the vector product \(\underline{a} \times \underline{b}\),
  3. the angle that \(\underline{a} \times \underline{b}\) makes with \(\underline{c}\).

Exercise 7

If position vectors \(\underline{OA}, \underline{OB}\) and \(\underline{OC}\) are given as \(2\underline{i} - \underline{j} + 3\underline{k}, 3\underline{i} + 2\underline{j} - 4\underline{k}\) and \(-\underline{i} + 2\underline{j} - 4\underline{k}\) respectively. Determine

  1. the vector \(\underline{AB}\),
  2. the vector \(\underline{BC}\),
  3. the vector product \(\underline{OA} \times \underline{OB}\),
  4. the area of the parallelogram with sides \(\underline{OA}\) and \(\underline{OB}\),
  5. the area of the triangle with sides \(\underline{OA}\) and \(\underline{OB}\),
  6. the angle between the vectors \(\underline{AB}\) and \(\underline{BC}\).

Exercise 8

If \(\underline{a} = 2\underline{i} - 3\underline{j} - \underline{k}\) and \(\underline{b} = \underline{i} + 4\underline{j} - 2\underline{k}\), determine

  1. \(\underline{a} \times \underline{b}\),
  2. \(\underline{b} \times \underline{a}\),
  3. \((\underline{a} + \underline{b}) \times (\underline{a} - \underline{b})\).

Exercise 9

If \(\underline{a} = 3\underline{i} - \underline{j} + 2\underline{k}, \underline{b} = 2\underline{i} + \underline{j} - \underline{k}\) and \(\underline{c} = \underline{i} - 2\underline{j} + 2\underline{k}\), determine

  1. \((\underline{a} \times \underline{b}) \times \underline{c}\),
  2. \(\underline{a} \times (\underline{b} \times \underline{c})\),
  3. \(5(\underline{a} \times \underline{b}), (5\underline{a})\times \underline{b}, \underline{a}\times(5\underline{b})\).

Exercise 10

(optional) Prove the sine rule using vectors.

Exercise 11

(optional) Prove the cosine rule using vectors.

Exercise 12

Show that the three vectors \(\underline{a} = 3\underline{i} - 2\underline{j} + \underline{k}, \underline{b} = \underline{i} - 3\underline{j} + 5\underline{k}\) and \(\underline{c}= 2\underline{i} + \underline{j} - 4\underline{k}\) form a right angled triangle.

Exercise 13

Determine the value of \(\alpha\) which will make the two vectors \(\underline{a} = 2\underline{i} + \alpha\underline{j} + \underline{k}\) and \(\underline{b} = 4\underline{i} - 2\underline{j} - 2\underline{k}\) be perpendicular to one another.

Exercise 14

Determine the area of the triangle having vertices at \(A=(1,3,2), B=(2, -1,1)\) and \(C=(-1,2,3)\).

Exercise 15

If \(\underline{a} = 3\underline{i} + 2\underline{j} - 4\underline{k}, \underline{b} = 4\underline{i} - 3\underline{j} + 2\underline{k}\) and \(\underline{c} = 2\underline{i} - 4\underline{j} - 3\underline{k}\) determine

  1. \(\underline{a} \cdot (\underline{b} \times \underline{c})\),
  2. \(\underline{b} \cdot (\underline{c} \times \underline{a})\),
  3. \(\underline{c} \cdot (\underline{a} \times \underline{b})\),
  4. \(\underline{a} \times (\underline{b} \times \underline{c})\),
  5. \(\underline{c} \times (\underline{a} \times \underline{b})\),
  6. \(\underline{b} \times (\underline{c} \times \underline{a})\).

Exercise 16

Given that a parallelepiped has its edges represented by the vectors \(\underline{a} = 4\underline{i} - 2\underline{j} - 3\underline{k}\), \(\underline{b} = \underline{i} + 2\underline{j} + \underline{k}\) and \(\underline{c} = 3\underline{i} - \underline{j} + 2\underline{k}\) determine the volume of the parallelepiped.

Exercise 17

Determine \(x\) so that the vectors \(\underline{a} = x\underline{i} - 2x\underline{j} - 3\underline{k}, \underline{b} =2 \underline{i} + \underline{j} - \underline{k}\) and \(\underline{c} = \underline{i} - \underline{j} + 14\underline{k}\) are coplanar.

[Solutions: 1. \(\underline{a}\cdot\underline{b}=8\), \(\underline{a}\times\underline{b} = 17\underline{i}-7\underline{j}+2\underline{k}\), angle \(67^{\circ}\); 2. (i) \(-7\), (ii) \(7\underline{i}-7\underline{j}-7\underline{k}\), (iii) \(120^{\circ}\); 3. (i) \(7\), \(5\underline{i}-3\underline{j}-\underline{k}\), (ii) \(8\), \(11\underline{i}+18\underline{j}-19\underline{k}\); 4. \(118^{\circ}\); 5. direction cosines are \(\frac{-4}{3\sqrt{20}}\), \(\frac{8}{3\sqrt{20}}\), \(\frac{10}{3\sqrt{20}}\); 6. (i) \(6\), \(82.7^{\circ}\), (ii) \(\frac{22}{47.1}\), \(\frac{-7}{47.1}\), \(\frac{-41}{47.1}\), (iii) \(19.6^{\circ}\); 7. (i) \(\underline{i}+3\underline{j}-7\underline{k}\), (ii) \(-4\underline{i}\), (iii) \(-2\underline{i}+17\underline{j}+7\underline{k}\), (iv) \(3\sqrt{38}\,\text{units}^2\), (v) \(3\sqrt{38}/2\,\text{units}^2\), (vi) \(\theta=97.5^{\circ}\); 8. (i) \(10\underline{i}+3\underline{j}+11\underline{k}\), (ii) \(-10\underline{i}-3\underline{j}-11\underline{k}\), (iii) \(-20\underline{i}-6\underline{j}-22\underline{k}\); 9. (i) \(24\underline{i}+7\underline{j}-5\underline{k}\), (ii) \(15\underline{i}+15\underline{j}-15\underline{k}\), (iii) \(-5\underline{i}+35\underline{j}+25\underline{k}\); 13. \(a=3\); 14. \(5.2\,\text{units}^2\); 15. (i) \(123\), (ii) \(123\), (iii) \(123\), (iv) \(44\underline{i}-38\underline{j}+14\underline{k}\), (v) \(2\underline{i}+58\underline{j}-54\underline{k}\), (vi) \(-46\underline{i} -20\underline{j}+62\underline{k}\); 16. \(3\,\text{units}^3\); 17. \(-9/71\);]