Question
For bold lower v equals 4 bold lower i minus 8 bold lower j, find unit vector u in the direction of v.
(1 point)
Responses
bold lower u equals one-third bold lower i minus two-thirds bold lower j
Image with alt text: bold lower u equals one-third bold lower i minus two-thirds bold lower j
bold lower u equals bold lower i minus bold lower j
Image with alt text: bold lower u equals bold lower i minus bold lower j
bold lower u equals start fraction start root 3 end root over 3 end fraction bold lower i minus start fraction 2 start root 3 end root over 3 end fraction bold lower j
Image with alt text: bold lower u equals start fraction start root 3 end root over 3 end fraction bold lower i minus start fraction 2 start root 3 end root over 3 end fraction bold lower j
bold lower u equals start fraction start root 5 end root over 5 end fraction bold lower i minus start fraction 2 start root 5 end root over 5 end fraction bold lower j
(1 point)
Responses
bold lower u equals one-third bold lower i minus two-thirds bold lower j
Image with alt text: bold lower u equals one-third bold lower i minus two-thirds bold lower j
bold lower u equals bold lower i minus bold lower j
Image with alt text: bold lower u equals bold lower i minus bold lower j
bold lower u equals start fraction start root 3 end root over 3 end fraction bold lower i minus start fraction 2 start root 3 end root over 3 end fraction bold lower j
Image with alt text: bold lower u equals start fraction start root 3 end root over 3 end fraction bold lower i minus start fraction 2 start root 3 end root over 3 end fraction bold lower j
bold lower u equals start fraction start root 5 end root over 5 end fraction bold lower i minus start fraction 2 start root 5 end root over 5 end fraction bold lower j
Answers
GPT-4o mini
To find the unit vector \( \mathbf{u} \) in the direction of the vector \( \mathbf{v} = 4 \mathbf{i} - 8 \mathbf{j} \), we follow these steps:
1. **Calculate the magnitude of \( \mathbf{v} \)**:
\[
\|\mathbf{v}\| = \sqrt{(4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}
\]
2. **Determine the unit vector \( \mathbf{u} \)**:
The unit vector in the direction of \( \mathbf{v} \) is given by:
\[
\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{4 \mathbf{i} - 8 \mathbf{j}}{4\sqrt{5}} = \frac{4}{4\sqrt{5}} \mathbf{i} - \frac{8}{4\sqrt{5}} \mathbf{j}
\]
Simplifying that:
\[
\mathbf{u} = \frac{1}{\sqrt{5}} \mathbf{i} - \frac{2}{\sqrt{5}} \mathbf{j}
\]
3. **Express in a more recognizable form**:
We can rewrite the unit vector \( \mathbf{u} \) as:
\[
\mathbf{u} = \frac{1}{\sqrt{5}} \mathbf{i} - \frac{2}{\sqrt{5}} \mathbf{j} = \frac{\sqrt{5}}{5} \mathbf{i} - \frac{2\sqrt{5}}{5} \mathbf{j}
\]
The correct expression for the unit vector \( \mathbf{u} \) derived from the calculations above matches the option:
\[
\mathbf{u} = \frac{\sqrt{5}}{5} \mathbf{i} - \frac{2\sqrt{5}}{5} \mathbf{j}
\]
Therefore, the answer is:
\[
\mathbf{u} = \frac{\sqrt{5}}{5} \mathbf{i} - \frac{2\sqrt{5}}{5} \mathbf{j}
\]
This corresponds to the option:
\[
\text{bold lower u equals } \frac{\sqrt{5}}{5} \bold{i} - \frac{2\sqrt{5}}{5} \bold{j}
\]
1. **Calculate the magnitude of \( \mathbf{v} \)**:
\[
\|\mathbf{v}\| = \sqrt{(4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}
\]
2. **Determine the unit vector \( \mathbf{u} \)**:
The unit vector in the direction of \( \mathbf{v} \) is given by:
\[
\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{4 \mathbf{i} - 8 \mathbf{j}}{4\sqrt{5}} = \frac{4}{4\sqrt{5}} \mathbf{i} - \frac{8}{4\sqrt{5}} \mathbf{j}
\]
Simplifying that:
\[
\mathbf{u} = \frac{1}{\sqrt{5}} \mathbf{i} - \frac{2}{\sqrt{5}} \mathbf{j}
\]
3. **Express in a more recognizable form**:
We can rewrite the unit vector \( \mathbf{u} \) as:
\[
\mathbf{u} = \frac{1}{\sqrt{5}} \mathbf{i} - \frac{2}{\sqrt{5}} \mathbf{j} = \frac{\sqrt{5}}{5} \mathbf{i} - \frac{2\sqrt{5}}{5} \mathbf{j}
\]
The correct expression for the unit vector \( \mathbf{u} \) derived from the calculations above matches the option:
\[
\mathbf{u} = \frac{\sqrt{5}}{5} \mathbf{i} - \frac{2\sqrt{5}}{5} \mathbf{j}
\]
Therefore, the answer is:
\[
\mathbf{u} = \frac{\sqrt{5}}{5} \mathbf{i} - \frac{2\sqrt{5}}{5} \mathbf{j}
\]
This corresponds to the option:
\[
\text{bold lower u equals } \frac{\sqrt{5}}{5} \bold{i} - \frac{2\sqrt{5}}{5} \bold{j}
\]