Complete parts (a) through (c) below. a. Find two unit vectors parallel to $$ \mathbf{v}=-24 i-7 \mathbf{j} $$. 0 of 1 b. Find $$ b $$ if $$ v=\left(\frac{1}{6}, b\right) $$ is a unit vector. c. Find all values of a such that $$ \mathbf{w}=\mathrm{ai}-\frac{\mathrm{a}}{10} \mathrm{j} $$ is a unit vector.

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Sure, let's tackle each part step by step.

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### **Part (a): Find Two Unit Vectors Parallel to $$ \mathbf{v} = -24\,\mathbf{i} - 7\,\mathbf{j} $$**

A **unit vector** is a vector with a magnitude of 1. To find unit vectors parallel to $$ \mathbf{v} $$, we first need to normalize $$ \mathbf{v} $$.

1. **Calculate the magnitude of $$ \mathbf{v} $$:**
   $$
   \|\mathbf{v}\| = \sqrt{(-24)^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25
   $$

2. **Normalize $$ \mathbf{v} $$ to get the first unit vector:**
   $$
   \mathbf{u}_1 = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{-24\,\mathbf{i} - 7\,\mathbf{j}}{25} = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j}
   $$

3. **Find the second unit vector by taking the negative of $$ \mathbf{u}_1 $$:**
   $$
   \mathbf{u}_2 = -\mathbf{u}_1 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j}
   $$

**Therefore, the two unit vectors parallel to $$ \mathbf{v} $$ are:**
$$
\mathbf{u}_1 = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j} \quad \text{and} \quad \mathbf{u}_2 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j}
$$

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### **Part (b): Find $$ b $$ if $$ \mathbf{v} = \left(\frac{1}{6}, b\right) $$ is a Unit Vector**

For $$ \mathbf{v} = \left(\frac{1}{6}, b\right) $$ to be a unit vector, its magnitude must be 1.

1. **Set up the equation for the magnitude:**
   $$
   \left\|\mathbf{v}\right\| = \sqrt{\left(\frac{1}{6}\right)^2 + b^2} = 1
   $$

2. **Solve for $$ b $$:**
   $$
   \sqrt{\frac{1}{36} + b^2} = 1 \
   \frac{1}{36} + b^2 = 1 \
   b^2 = 1 - \frac{1}{36} \
   b^2 = \frac{36}{36} - \frac{1}{36} \
   b^2 = \frac{35}{36} \
   b = \pm \frac{\sqrt{35}}{6}
   $$

**Therefore, the possible values of $$ b $$ are:**
$$
b = \frac{\sqrt{35}}{6} \quad \text{or} \quad b = -\frac{\sqrt{35}}{6}
$$

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### **Part (c): Find All Values of $$ a $$ Such That $$ \mathbf{w} = a\,\mathbf{i} - \frac{a}{10}\,\mathbf{j} $$ is a Unit Vector**

For $$ \mathbf{w} = a\,\mathbf{i} - \frac{a}{10}\,\mathbf{j} $$ to be a unit vector, its magnitude must be 1.

1. **Express the magnitude of $$ \mathbf{w} $$:**
   $$
   \left\|\mathbf{w}\right\| = \sqrt{a^2 + \left(-\frac{a}{10}\right)^2} = \sqrt{a^2 + \frac{a^2}{100}} = \sqrt{\frac{100a^2 + a^2}{100}} = \sqrt{\frac{101a^2}{100}} = \frac{\sqrt{101}\,|a|}{10}
   $$

2. **Set the magnitude equal to 1 and solve for $$ a $$:**
   $$
   \frac{\sqrt{101}\,|a|}{10} = 1 \
   |a| = \frac{10}{\sqrt{101}} \
   a = \pm \frac{10}{\sqrt{101}}
   $$

To rationalize the denominator:
   $$
   a = \pm \frac{10\sqrt{101}}{101}
   $$

**Therefore, the values of $$ a $$ are:**
$$
a = \frac{10\sqrt{101}}{101} \quad \text{and} \quad a = -\frac{10\sqrt{101}}{101}
$$

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**Summary of Answers:**

- **(a)** The two unit vectors parallel to $$ \mathbf{v} = -24\,\mathbf{i} -7\,\mathbf{j} $$ are:
  $$
  \mathbf{u}_1 = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j}, \quad \mathbf{u}_2 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j}
  $$

- **(b)** The values of $$ b $$ are:
  $$
  b = \pm \frac{\sqrt{35}}{6}
  $$

- **(c)** The values of $$ a $$ are:
  $$
  a = \pm \frac{10\sqrt{101}}{101}
  $$
**Part (a):** The two unit vectors parallel to $$ \mathbf{v} = -24\,\mathbf{i} -7\,\mathbf{j} $$ are: $$ \mathbf{u}_1 = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j} \quad \text{and} \quad \mathbf{u}_2 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j} $$ **Part (b):** The values of $$ b $$ are: $$ b = \frac{\sqrt{35}}{6} \quad \text{and} \quad b = -\frac{\sqrt{35}}{6} $$ **Part (c):** The values of $$ a $$ are: $$ a = \frac{10\sqrt{101}}{101} \quad \text{and} \quad a = -\frac{10\sqrt{101}}{101} $$

Complete parts (a) through © below.
a. Find two unit vectors parallel to . 0 of 1
b. Find if is a unit vector.
c. Find all values of a such that is a unit vector.

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Solución

Part (a): Find Two Unit Vectors Parallel to

Part (b): Find if is a Unit Vector

Part ©: Find All Values of Such That is a Unit Vector

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Complete parts (a) through © below. a. Find two unit vectors parallel to . 0 of 1 b. Find if is a unit vector. c. Find all values of a such that is a unit vector.