Complete parts (a) through (c) below. a. Find two unit vectors parallel to $$ \mathbf{v}=-24 i-7 \mathbf{j} $$. b. Find $$ b $$ if $$ v=\left\langle\frac{1}{2}, \mathrm{~b}\right\rangle $$ is a unit vector. c. Find all values of a such that $$ w=a i-\frac{a}{5} j $$ is a unit vector. a. The parallel unit vector with the same direction is

Question

UpStudy AI · Accepted Answer

To solve the problem, we will go through each part step by step.

### Part (a)
We need to find two unit vectors parallel to $$ \mathbf{v} = -24 \mathbf{i} - 7 \mathbf{j} $$.

1. **Calculate 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 direction of $$ \mathbf{v} $$**:
   The unit vector $$ \mathbf{u} $$ in the direction of $$ \mathbf{v} $$ is given by:
   $$
   \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{-24 \mathbf{i} - 7 \mathbf{j}}{25} = -\frac{24}{25} \mathbf{i} - \frac{7}{25} \mathbf{j}
   $$

3. **Find the unit vector in the opposite direction**:
   The unit vector in the opposite direction is:
   $$
   -\mathbf{u} = \frac{24}{25} \mathbf{i} + \frac{7}{25} \mathbf{j}
   $$

Thus, the two unit vectors parallel to $$ \mathbf{v} $$ are:
1. $$ -\frac{24}{25} \mathbf{i} - \frac{7}{25} \mathbf{j} $$
2. $$ \frac{24}{25} \mathbf{i} + \frac{7}{25} \mathbf{j} $$

### Part (b)
We need to find $$ b $$ if $$ \mathbf{v} = \left\langle \frac{1}{2}, b \right\rangle $$ is a unit vector.

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

2. **Square both sides**:
   $$
   \left(\frac{1}{2}\right)^2 + b^2 = 1
   $$
   $$
   \frac{1}{4} + b^2 = 1
   $$

3. **Solve for $$ b^2 $$**:
   $$
   b^2 = 1 - \frac{1}{4} = \frac{3}{4}
   $$

4. **Find $$ b $$**:
   $$
   b = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}
   $$

Thus, the values of $$ b $$ are $$ \frac{\sqrt{3}}{2} $$ and $$ -\frac{\sqrt{3}}{2} $$.

### Part (c)
We need to find all values of $$ a $$ such that $$ \mathbf{w} = a \mathbf{i} - \frac{a}{5} \mathbf{j} $$ is a unit vector.

1. **Set up the equation for the magnitude of $$ \mathbf{w} $$**:
   $$
   \|\mathbf{w}\| = \sqrt{a^2 + \left(-\frac{a}{5}\right)^2} = 1
   $$

2. **Square both sides**:
   $$
   a^2 + \left(-\frac{a}{5}\right)^2 = 1
   $$
   $$
   a^2 + \frac{a^2}{25} = 1
   $$

3. **Combine the terms**:
   $$
   a^2 \left(1 + \frac{1}{25}\right) = 1
   $$
   $$
   a^2 \left(\frac{25 + 1}{25}\right) = 1
   $$
   $$
   a^2 \left(\frac{26}{25}\right) = 1
   $$

4. **Solve for $$ a^2 $$**:
   $$
   a^2 = \frac{25}{26}
   $$

5. **Find $$ a $$**:
   $$
   a = \pm \sqrt{\frac{25}{26}} = \pm \frac{5}{\sqrt{26}} = \pm \frac{5\sqrt{26}}{26}
   $$

Thus, the values of $$ a $$ are $$ \frac{5\sqrt{26}}{26} $$ and $$ -\frac{5\sqrt{26}}{26} $$.

### Summary of Answers
- **Part (a)**: The two unit vectors are $$ -\frac{24}{25} \mathbf{i} - \frac{7}{25} \mathbf{j} $$ and $$ \frac{24}{25} \mathbf{i} + \frac{7}{25} \mathbf{j} $$.
- **Part (b)**: The values of $$ b $$ are $$ \frac{\sqrt{3}}{2} $$ and $$ -\frac{\sqrt{3}}{2} $$.
- **Part (c)**: The values of $$ a $$ are $$ \frac{5\sqrt{26}}{26} $$ and $$ -\frac{5\sqrt{26}}{26} $$.
- **Part (a)**: The two unit vectors are $$ -\frac{24}{25} \mathbf{i} - \frac{7}{25} \mathbf{j} $$ and $$ \frac{24}{25} \mathbf{i} + \frac{7}{25} \mathbf{j} $$. - **Part (b)**: $$ b = \frac{\sqrt{3}}{2} $$ or $$ b = -\frac{\sqrt{3}}{2} $$. - **Part (c)**: $$ a = \frac{5\sqrt{26}}{26} $$ or $$ a = -\frac{5\sqrt{26}}{26} $$.

Solución de tutoría real

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Pre Calculus Questions