The problem asks to find several vector projections given the vectors $u = i + 2j$, $v = 2i - j$, and $w = i + 5j$. Specifically, we need to compute the following projections: 23. $proj_v u$ 24. $proj_u v$ 25. $proj_u w$ 26. $proj_u (w - v)$ 27. $proj_j u$ 28. $proj_i u$

1. Problem Description

The problem asks to find several vector projections given the vectors $u = i + 2j$, $v = 2i - j$, and $w = i + 5j$. Specifically, we need to compute the following projections:

2

3. $proj_v u$

2

4. $proj_u v$

2

5. $proj_u w$

2

6. $proj_u (w - v)$

2

7. $proj_j u$

2

8. $proj_i u$

2. Solution Steps

The formula for the projection of vector $a$ onto vector $b$ is given by:

$proj_b a = \frac{a \cdot b}{||b||^2} b$

where $a \cdot b$ is the dot product of $a$ and $b$, and $||b||^2$ is the square of the magnitude of $b$.

We have the following vectors:

$u = i + 2j = <1, 2>$

$v = 2i - j = <2, -1>$

$w = i + 5j = <1, 5>$

$i = <1, 0>$

$j = <0, 1>$

2

3. $proj_v u$

$u \cdot v = (1)(2) + (2)(-1) = 2 - 2 = 0$

$||v||^2 = (2)^2 + (-1)^2 = 4 + 1 = 5$

$proj_v u = \frac{u \cdot v}{||v||^2} v = \frac{0}{5} v = 0 <2, -1> = <0, 0> = 0i + 0j$

2

4. $proj_u v$

$v \cdot u = (2)(1) + (-1)(2) = 2 - 2 = 0$

$||u||^2 = (1)^2 + (2)^2 = 1 + 4 = 5$

$proj_u v = \frac{v \cdot u}{||u||^2} u = \frac{0}{5} u = 0 <1, 2> = <0, 0> = 0i + 0j$

2

5. $proj_u w$

$w \cdot u = (1)(1) + (5)(2) = 1 + 10 = 11$

$||u||^2 = (1)^2 + (2)^2 = 1 + 4 = 5$

$proj_u w = \frac{w \cdot u}{||u||^2} u = \frac{11}{5} <1, 2> = <\frac{11}{5}, \frac{22}{5}> = \frac{11}{5}i + \frac{22}{5}j$

2

6. $proj_u (w - v)$

$w - v = <1, 5> - <2, -1> = <1-2, 5-(-1)> = <-1, 6>$

$(w - v) \cdot u = (-1)(1) + (6)(2) = -1 + 12 = 11$

$||u||^2 = (1)^2 + (2)^2 = 1 + 4 = 5$

$proj_u (w - v) = \frac{(w - v) \cdot u}{||u||^2} u = \frac{11}{5} <1, 2> = <\frac{11}{5}, \frac{22}{5}> = \frac{11}{5}i + \frac{22}{5}j$

2

7. $proj_j u$

$u \cdot j = (1)(0) + (2)(1) = 0 + 2 = 2$

$||j||^2 = (0)^2 + (1)^2 = 0 + 1 = 1$

$proj_j u = \frac{u \cdot j}{||j||^2} j = \frac{2}{1} <0, 1> = 2 <0, 1> = <0, 2> = 0i + 2j$

2

8. $proj_i u$

$u \cdot i = (1)(1) + (2)(0) = 1 + 0 = 1$

$||i||^2 = (1)^2 + (0)^2 = 1 + 0 = 1$

$proj_i u = \frac{u \cdot i}{||i||^2} i = \frac{1}{1} <1, 0> = 1 <1, 0> = <1, 0> = 1i + 0j$

3. Final Answer

2

3. $proj_v u = 0i + 0j$

2

4. $proj_u v = 0i + 0j$

2

5. $proj_u w = \frac{11}{5}i + \frac{22}{5}j$

2

6. $proj_u (w - v) = \frac{11}{5}i + \frac{22}{5}j$

2

7. $proj_j u = 0i + 2j$

2

8. $proj_i u = 1i + 0j$