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

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### **a. Find two unit vectors parallel to $$ \mathbf{v} = -24\,\mathbf{i} - 7\,\mathbf{j} $$.**

To find unit vectors parallel to $$ \mathbf{v} $$, we need to:

1. **Determine the magnitude of $$ \mathbf{v} $$:**
   
   $$
   \|\mathbf{v}\| = \sqrt{(-24)^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25
   $$
   
2. **Find the unit vector in the same direction as $$ \mathbf{v} $$:**
   
   $$
   \mathbf{u}_1 = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{-24\,\mathbf{i} - 7\,\mathbf{j}}{25} = \left( -\frac{24}{25} \right)\mathbf{i} + \left( -\frac{7}{25} \right)\mathbf{j}
   $$
   
3. **Find the unit vector in the opposite direction of $$ \mathbf{v} $$:**
   
   $$
   \mathbf{u}_2 = -\mathbf{u}_1 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j}
   $$
   
**Answer:**
$$
\mathbf{u}_1 = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j} \quad \text{and} \quad \mathbf{u}_2 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j}
$$

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### **b. Find $$ b $$ if $$ \mathbf{v} = \left\langle \frac{1}{6}, b \right\rangle $$ is a unit vector.**

A unit vector has a magnitude of 1. Therefore,

$$
\|\mathbf{v}\| = \sqrt{\left(\frac{1}{6}\right)^2 + b^2} = 1
$$

Squaring both sides to eliminate the square root:

$$
\left( \frac{1}{6} \right)^2 + b^2 = 1 \
\frac{1}{36} + b^2 = 1 \
b^2 = 1 - \frac{1}{36} \
b^2 = \frac{35}{36}
$$

Taking the square root of both sides:

$$
b = \pm \frac{\sqrt{35}}{6}
$$

**Answer:**
$$
b = \frac{\sqrt{35}}{6} \quad \text{or} \quad b = -\frac{\sqrt{35}}{6}
$$

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### **c. Find all values of $$ a $$ such that $$ \mathbf{w} = a\,\mathbf{i} - \frac{a}{10}\,\mathbf{j} $$ is a unit vector.**

Again, requiring the magnitude of $$ \mathbf{w} $$ to be 1:

$$
\|\mathbf{w}\| = \sqrt{a^2 + \left( \frac{a}{10} \right)^2} = 1
$$

Squaring both sides:

$$
a^2 + \frac{a^2}{100} = 1 \
\frac{100a^2 + a^2}{100} = 1 \
101a^2 = 100 \
a^2 = \frac{100}{101}
$$

Taking the square root:

$$
a = \pm \frac{10}{\sqrt{101}} = \pm \frac{10\sqrt{101}}{101}
$$

**Answer:**
$$
a = \frac{10}{\sqrt{101}} \quad \text{or} \quad a = -\frac{10}{\sqrt{101}}
$$

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Each of these solutions ensures that the resulting vectors are indeed unit vectors by satisfying the condition $$ \|\mathbf{v}\| = 1 $$.
a. The two unit vectors parallel to $$ \mathbf{v} = -24\,\mathbf{i} - 7\,\mathbf{j} $$ are: $$ \mathbf{u}_1 = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j} \quad \text{and} \quad \mathbf{u}_2 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j} $$ b. The value of $$ b $$ is: $$ b = \frac{\sqrt{35}}{6} \quad \text{or} \quad b = -\frac{\sqrt{35}}{6} $$ c. The values of $$ a $$ are: $$ a = \frac{10}{\sqrt{101}} \quad \text{or} \quad a = -\frac{10}{\sqrt{101}} $$

Complete parts (a) through © below.
a. Find two unit vectors parallel to .
b. Find b if is a unit vector.
c. Find all values of a such that is a unit vector.
a. The parallel unit vector with the same direction is

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Complete parts (a) through © below. a. Find two unit vectors parallel to . b. Find b if is a unit vector. c. Find all values of a such that is a unit vector. a. The parallel unit vector with the same direction is