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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”.

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. $(-1+3e_1+2e_2)+0=-1+3e_1+2e_2$