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author: niplav, created: 2024-08-06, modified: 2024-09-10, language: english, status: in progress, importance: 2, confidence: likely

Solutions to the video series “From Zero to Geo”.

Contents

Solutions to “From Zero to Geo”

1.1

  1. Same length: {red vector, blue vector, white vector}, {green vector, orange vector, yellow vector}, {grey vector, pink vector}
  2. Same orientation: {blue vector, green vector}, {yellow vector, orange vector}, {grey vector, white vector}
  3. Equal vectors: only {yellow vector, orange vector}

1.2

1

  1. Length of \overset{\rightarrow}{v}_1 : 3
  2. Length of \overset{\rightarrow}{v}_2 : 2
  3. Length of \overset{\rightarrow}{v}_3 : 1
  4. Length of \overset{\rightarrow}{v}_4 : 5
  5. Length of \overset{\rightarrow}{v}_5 : 2\sqrt{2}

2

  1. 2 \overset{\rightarrow}{v}_1 : [6, 0]
  2. \overset{\rightarrow}{v}_2 : [0, -1]
  3. \overset{\rightarrow}{v}_3 : [3, 0]
  4. \overset{\rightarrow}{v}_4 : \overset{\rightarrow}{0}
  5. \overset{\rightarrow}{v}_5 : [-3, -3]

1.3

1

  1. \overset{\rightarrow}{v}_1+\overset{\rightarrow}{v}_1=[6, 0]
  2. \overset{\rightarrow}{v}_2+\overset{\rightarrow}{v}_3=[1, -2]
  3. \overset{\rightarrow}{v}_5+\overset{\rightarrow}{v}_1=[-3, -2]
  4. \overset{\rightarrow}{v}_4+3\overset{\rightarrow}{v}_3=[0, -4]
  5. 3\overset{\rightarrow}{v}_3+\overset{\rightarrow}{v}_1=[0, 0]

2

Subtraction: \overset{\rightarrow}{v}_1-\overset{\rightarrow}{v}_2=\overset{\rightarrow}{v}_1+-1 \cdot \overset{\rightarrow}{v}_2 .

Geometrically: Turn around \overset{\rightarrow}{v}_2 , then walk along \overset{\rightarrow}{v}_1 and then along the turned \overset{\rightarrow}{v}_2 .

3

  1. \overset{\rightarrow}{v}_1 : [-3, 0]
  2. \overset{\rightarrow}{v}_2 : [0, 1]
  3. \overset{\rightarrow}{v}_3 : [-1, 0]
  4. \overset{\rightarrow}{v}_4 : [-3, 4]
  5. \overset{\rightarrow}{v}_5 : [2, 2]

4

  1. \overset{\rightarrow}{v}_1-\overset{\rightarrow}{v}_2=[3, 2]
  2. \overset{\rightarrow}{v}_5-\overset{\rightarrow}{v}_3=[-1, -2]
  3. \overset{\rightarrow}{v}_4-\overset{\rightarrow}{v}_1=[0, -4]
  4. \overset{\rightarrow}{v}_1-\overset{\rightarrow}{v}_3=[4, 0]
  5. \overset{\rightarrow}{v}_4-\overset{\rightarrow}{v}_4=[0, 0]

1.11

1

  1. (1+e_1+e_2)+(2+e_1+3e_2)=3+2e_1+4e_2
  2. (-3+2e_1-e_2)+(-8-7e_1-6e_2)=5-5e_1-7e_2
  3. (4-3e_1-5e_2)+(-4+3e_1-5e_2)=0
  4. (2)+(-1+3e_1-2e_2)=1+3e_1+2e_2
  5. (-1+e_2)+(-3e_1+3e_2)=-1-3e_1+4e_2
  6. (-1+3e_1+2e_2)+0=-1+3e_1+2e_2