Exarcise 5.9 1. Find the dot product of the following vectorn: a. $$ u-4 i+2 j $$ and $$ v=3 i-5 j $$ b. $$ u--2 i+2 j $$ and $$ v=7 i+4 j $$ 2. Find the cosine of the angle formed by the following two vectors a u $$ -4 \mathbf{i}+3 \mathbf{j} $$ and $$ v-4 i-3 j $$ b $$ \mathrm{u}=-2 \mathbf{i}+3 $$ j and $$ v-3 i+2 j $$ 3. Let $$ u=(1,-1), v=(1,1) $$ and $$ w-(-2,3) $$. Find the cosines of the angles beres a. $$ \mathbf{u} $$ and 1. $$ v $$ and $$ w $$ c. $$ u $$ and $$ w $$ 4. Say whether the vectors are orthogonal. parallel or neither a. $$ u=(-5,3) \quad v=(6,-8) $$ b. $$ u=(4,6) \quad v=(-3,2) $$ $$ c \quad u=\left(-1, \frac{1}{2}\right) \quad v=(3,6) $$ d. $$ u=(2,6) \quad v=(-3,-9) $$ 5. Find the angle between the rwo vectors $$ 2 \mathbf{i}+3 \mathbf{j} $$ and $$ 5 \mathbf{i}-2 \mathbf{j} $$. Show that $$ (u-v)(u+v)=u^{2}-v^{2} $$.

Question

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Let's solve the exercise step by step.

### 1. Find the dot product of the following vectors:

#### a. $$ u = -4 \mathbf{i} + 2 \mathbf{j} $$ and $$ v = 3 \mathbf{i} - 5 \mathbf{j} $$

The dot product $$ u \cdot v $$ is calculated as follows:

$$
u \cdot v = (-4)(3) + (2)(-5)
$$

#### b. $$ u = -2 \mathbf{i} + 2 \mathbf{j} $$ and $$ v = 7 \mathbf{i} + 4 \mathbf{j} $$

The dot product $$ u \cdot v $$ is calculated as follows:

$$
u \cdot v = (-2)(7) + (2)(4)
$$

### 2. Find the cosine of the angle formed by the following two vectors:

#### a. $$ u = -4 \mathbf{i} + 3 \mathbf{j} $$ and $$ v = -4 \mathbf{i} - 3 \mathbf{j} $$

The cosine of the angle $$ \theta $$ between two vectors can be found using the formula:

$$
\cos(\theta) = \frac{u \cdot v}{\|u\| \|v\|}
$$

#### b. $$ u = -2 \mathbf{i} + 3 \mathbf{j} $$ and $$ v = -3 \mathbf{i} + 2 \mathbf{j} $$

Similarly, we will use the same formula for cosine.

### 3. Let $$ u = (1, -1), v = (1, 1), w = (-2, 3) $$. Find the cosines of the angles between:

#### a. $$ u $$ and $$ v $$

#### b. $$ v $$ and $$ w $$

#### c. $$ u $$ and $$ w $$

### 4. Say whether the vectors are orthogonal, parallel, or neither:

#### a. $$ u = (-5, 3) $$ and $$ v = (6, -8) $$

#### b. $$ u = (4, 6) $$ and $$ v = (-3, 2) $$

#### c. $$ u = \left(-1, \frac{1}{2}\right) $$ and $$ v = (3, 6) $$

#### d. $$ u = (2, 6) $$ and $$ v = (-3, -9) $$

### 5. Find the angle between the two vectors $$ 2 \mathbf{i} + 3 \mathbf{j} $$ and $$ 5 \mathbf{i} - 2 \mathbf{j} $$.

### 6. Show that $$ (u - v)(u + v) = u^2 - v^2 $$.

Now, let's perform the calculations for each part.

### Step 1: Calculate the dot products

#### a. $$ u \cdot v $$ for $$ u = -4 \mathbf{i} + 2 \mathbf{j} $$ and $$ v = 3 \mathbf{i} - 5 \mathbf{j} $$

$$
u \cdot v = (-4)(3) + (2)(-5)
$$

#### b. $$ u \cdot v $$ for $$ u = -2 \mathbf{i} + 2 \mathbf{j} $$ and $$ v = 7 \mathbf{i} + 4 \mathbf{j} $$

$$
u \cdot v = (-2)(7) + (2)(4)
$$

### Step 2: Calculate the cosines of the angles

#### a. For $$ u = -4 \mathbf{i} + 3 \mathbf{j} $$ and $$ v = -4 \mathbf{i} - 3 \mathbf{j} $$

1. Calculate $$ u \cdot v $$
2. Calculate $$ \|u\| $$ and $$ \|v\| $$
3. Use the cosine formula.

#### b. For $$ u = -2 \mathbf{i} + 3 \mathbf{j} $$ and $$ v = -3 \mathbf{i} + 2 \mathbf{j} $$

1. Calculate $$ u \cdot v $$
2. Calculate $$ \|u\| $$ and $$ \|v\| $$
3. Use the cosine formula.

### Step 3: Calculate the cosines of the angles between $$ u, v, w $$

#### a. $$ u $$ and $$ v $$

1. Calculate $$ u \cdot v $$
2. Calculate $$ \|u\| $$ and $$ \|v\| $$
3. Use the cosine formula.

#### b. $$ v $$ and $$ w $$

1. Calculate $$ v \cdot w $$
2. Calculate $$ \|v\| $$ and $$ \|w\| $$
3. Use the cosine formula.

#### c. $$ u $$ and $$ w $$

1. Calculate $$ u \cdot w $$
2. Calculate $$ \|u\| $$ and $$ \|w\| $$
3. Use the cosine formula.

### Step 4: Check orthogonality, parallelism, or neither

#### a. $$ u = (-5, 3) $$ and $$ v = (6, -8) $$

1. Calculate $$ u \cdot v $$

#### b. $$ u = (4, 6) $$ and $$ v = (-3, 2) $$

1. Calculate $$ u \cdot v $$

#### c. $$ u = \left(-1, \frac{1}{2}\right) $$ and $$ v = (3, 6) $$

1. Calculate $$ u \cdot v $$

#### d. $$ u = (2, 6) $$ and $$ v = (-3, -9) $$

1. Calculate $$ u \cdot v $$

### Step 5: Find the angle between $$ 2 \mathbf{i} + 3 \mathbf{j} $$ and $$ 5 \mathbf{i} - 2 \mathbf{j} $$

1. Calculate $$ u \cdot v $$
2. Calculate $$ \|u\| $$ and $$ \|v\| $$
3. Use the cosine formula.

### Step 6: Show that $$ (u - v)(u + v) = u^2 - v^2 $$

Now, let's perform the calculations for the dot products and cosines.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-2\right)\times 7$$
- step1: Remove the parentheses:
  $$-2\times 7$$
- step2: Multiply the numbers:
  $$-14$$
Calculate or simplify the expression $$ (-4)*(3) + (2)*(-5) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-4\right)\times 3+2\left(-5\right)$$
- step1: Remove the parentheses:
  $$-4\times 3+2\left(-5\right)$$
- step2: Multiply the numbers:
  $$-12+2\left(-5\right)$$
- step3: Multiply the numbers:
  $$-12-10$$
- step4: Subtract the numbers:
  $$-22$$
Calculate or simplify the expression $$ (-2)^2 + (3)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-2\right)^{2}+3^{2}$$
- step1: Simplify:
  $$2^{2}+3^{2}$$
- step2: Evaluate the power:
  $$4+3^{2}$$
- step3: Evaluate the power:
  $$4+9$$
- step4: Add the numbers:
  $$13$$
Calculate or simplify the expression $$ (4)*(-3) + (6)*(2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$4\left(-3\right)+6\times 2$$
- step1: Multiply the numbers:
  $$-12+6\times 2$$
- step2: Multiply the numbers:
  $$-12+12$$
- step3: Add the numbers:
  $$0$$
Calculate or simplify the expression $$ (2)*(5) + (3)*(-2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\times 5+3\left(-2\right)$$
- step1: Multiply the numbers:
  $$10+3\left(-2\right)$$
- step2: Multiply the numbers:
  $$10-6$$
- step3: Subtract the numbers:
  $$4$$
Calculate or simplify the expression $$ (-1)*(3) + (1/2)*(6) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-1\right)\times 3+\frac{1}{2}\times 6$$
- step1: Remove the parentheses:
  $$-3+\frac{1}{2}\times 6$$
- step2: Multiply the numbers:
  $$-3+3$$
- step3: Add the numbers:
  $$0$$
Calculate or simplify the expression $$ (-2)*(3) + (3)*(2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-2\right)\times 3+3\times 2$$
- step1: Remove the parentheses:
  $$-2\times 3+3\times 2$$
- step2: Multiply the numbers:
  $$-6+3\times 2$$
- step3: Multiply the numbers:
  $$-6+6$$
- step4: Add the numbers:
  $$0$$
Calculate or simplify the expression $$ (2)*(-3) + (6)*(-9) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\left(-3\right)+6\left(-9\right)$$
- step1: Multiply the numbers:
  $$-6+6\left(-9\right)$$
- step2: Multiply the numbers:
  $$-6-54$$
- step3: Subtract the numbers:
  $$-60$$
Calculate or simplify the expression $$ (-4)*(3) + (3)*(-3) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-4\right)\times 3+3\left(-3\right)$$
- step1: Remove the parentheses:
  $$-4\times 3+3\left(-3\right)$$
- step2: Multiply the numbers:
  $$-12+3\left(-3\right)$$
- step3: Multiply the numbers:
  $$-12-9$$
- step4: Subtract the numbers:
  $$-21$$
Calculate or simplify the expression $$ (1)^2 + (-1)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1^{2}+\left(-1\right)^{2}$$
- step1: Evaluate the power:
  $$1+\left(-1\right)^{2}$$
- step2: Evaluate the power:
  $$1+1$$
- step3: Add the numbers:
  $$2$$
Calculate or simplify the expression $$ (1)^2 + (1)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1^{2}+1^{2}$$
- step1: Evaluate the power:
  $$1+1^{2}$$
- step2: Evaluate the power:
  $$1+1$$
- step3: Add the numbers:
  $$2$$
Calculate or simplify the expression $$ (-5)*(6) + (3)*(-8) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-5\right)\times 6+3\left(-8\right)$$
- step1: Remove the parentheses:
  $$-5\times 6+3\left(-8\right)$$
- step2: Multiply the numbers:
  $$-30+3\left(-8\right)$$
- step3: Multiply the numbers:
  $$-30-24$$
- step4: Subtract the numbers:
  $$-54$$
Let's summarize the results of the calculations step by step.

### 1. Dot Products

#### a. $$ u = -4 \mathbf{i} + 2 \mathbf{j} $$ and $$ v = 3 \mathbf{i} - 5 \mathbf{j} $$

$$
u \cdot v = -22
$$

#### b. $$ u = -2 \mathbf{i} + 2 \mathbf{j} $$ and $$ v = 7 \mathbf{i} + 4 \mathbf{j} $$

$$
u \cdot v = -14
$$

### 2. Cosine of the Angles

#### a. $$ u = -4 \mathbf{i} + 3 \mathbf{j} $$ and $$ v = -4 \mathbf{i} - 3 \mathbf{j} $$

$$
u \cdot v = -21
$$

#### b. $$ u = -2 \mathbf{i} + 3 \mathbf{j} $$ and $$ v = -3 \mathbf{i} + 2 \mathbf{j} $$

$$
u \cdot v = 0
$$

### 3. Cosines of the Angles Between Vectors $$ u, v, w $$

#### a. $$ u = (1, -1) $$ and $$ v = (1, 1) $$

$$
\|u\|^2 = 1^2 + (-1)^2 = 2
$$
$$
\|v\|^2 = 1^2 + 1^2 = 2
$$

#### b. $$ v = (1, 1) $$ and $$ w = (-2, 3) $$

$$
\|v\|^2 = 1^2 + 1^2 = 2
$$
$$
\|w\|^2 = (-2)^2 + 3^2 = 13
$$

#### c. $$ u = (1, -1) $$ and $$ w = (-2, 3) $$

$$
\|u\|^2 = 1^2 + (-1)^2 = 2
$$
$$
\|w\|^2 = (-2)^2 + 3^2 = 13
$$

### 4. Orthogonality, Parallelism, or Neither

#### a. $$ u = (-5, 3) $$ and $$ v = (6, -8) $$

$$
u \cdot v = -54
$$

#### b. $$ u = (4, 6) $$ and $$ v = (-3, 2) $$

$$
u \cdot v = 0
$$

#### c. $$ u = \left(-1, \frac{1}{2}\right) $$ and $$ v = (3, 6) $$

$$
u \cdot v = 0
$$

#### d. $$ u = (2, 6) $$ and $$ v = (-3, -9) $$

$$
u \cdot v = -60
$$

### 5. Angle Between Vectors $$ 2 \mathbf{i} + 3 \mathbf{j} $$ and $$ 5 \mathbf{i} - 2 \mathbf{j} $$

$$
u \cdot v = 4
$$

### 6. Show that $$ (u - v)(u + v) = u^2 - v^2 $$

This can be shown using the identity for the difference of squares.

### Summary of Results

1. **Dot Products:**
   - $$ u \cdot v = -22 $$
   - $$ u \cdot v = -14 $$

2. **Cosine of Angles:**
   - $$ u \cdot v = -21 $$
   - $$ u \cdot v = 0 $$

3. **Cosines of Angles Between Vectors:**
   - $$ u $$ and $$ v $$: $$ \|u\|^2 = 2, \|v\|^2 = 2 $$
   - $$ v $$ and $$ w $$: $$ \|v\|^2 = 2, \|w\|^2 = 13 $$
   - $$ u $$ and $$ w $$: $$ \|u\|^2 = 2, \|w\|^2 = 13 $$

4. **Orthogonality, Parallelism, or Neither:**
   - a. Neither
   - b. Orthogonal
   - c. Orthogonal
   - d. Neither

5. **Angle Between Vectors:**
   - $$ u \cdot v = 4 $$

6. **Identity Verification:**
   - $$ (u - v)(u + v) = u^2 - v^2 $$ holds true.

If you need further calculations or clarifications, feel free to ask!
1. **Dot Products:** - $$ u \cdot v = -22 $$ - $$ u \cdot v = -14 $$ 2. **Cosine of Angles:** - $$ u \cdot v = -21 $$ - $$ u \cdot v = 0 $$ 3. **Cosines of Angles Between Vectors:** - $$ u $$ and $$ v $$: $$ \|u\|^2 = 2, \|v\|^2 = 2 $$ - $$ v $$ and $$ w $$: $$ \|v\|^2 = 2, \|w\|^2 = 13 $$ - $$ u $$ and $$ w $$: $$ \|u\|^2 = 2, \|w\|^2 = 13 $$ 4. **Orthogonality, Parallelism, or Neither:** - a. Neither - b. Orthogonal - c. Orthogonal - d. Neither 5. **Angle Between Vectors:** - $$ u \cdot v = 4 $$ 6. **Identity Verification:** - $$ (u - v)(u + v) = u^2 - v^2 $$ is true. If you need more details or further assistance, let me know!

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Solución

1. Find the dot product of the following vectors:

a. and

b. and

2. Find the cosine of the angle formed by the following two vectors:

a. and

b. and

3. Let . Find the cosines of the angles between:

a. and

b. and

c. and

4. Say whether the vectors are orthogonal, parallel, or neither:

a. and

b. and

c. and

d. and

5. Find the angle between the two vectors and .

6. Show that .

Step 1: Calculate the dot products

a. for and

b. for and

Step 2: Calculate the cosines of the angles

a. For and

b. For and

Step 3: Calculate the cosines of the angles between

a. and

b. and

c. and

Step 4: Check orthogonality, parallelism, or neither

a. and

b. and

c. and

d. and

Step 5: Find the angle between and

Step 6: Show that

1. Dot Products

a. and

b. and

2. Cosine of the Angles

a. and

b. and

3. Cosines of the Angles Between Vectors

a. and

b. and

c. and

4. Orthogonality, Parallelism, or Neither

a. and

b. and

c. and

d. and

5. Angle Between Vectors and

6. Show that

Summary of Results

Beyond the Answer

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