find the component form of the unit vector that makes the angle -3pi/4 with a positive x axis

To find the component form of a unit vector that makes an angle \(-\frac{3\pi}{4}\) with the positive \(x\)-axis, we can use the following general formula for the unit vector in two dimensions:

\[
\mathbf{v} = \langle \cos(\theta), \sin(\theta) \rangle
\]

Where:
- \(\theta\) is the angle with the positive \(x\)-axis.
- Since we want a unit vector, the magnitude will always be 1.

Given that \(\theta = -\frac{3\pi}{4}\), we plug it into the formula:

\[
\mathbf{v} = \langle \cos\left(-\frac{3\pi}{4}\right), \sin\left(-\frac{3\pi}{4}\right) \rangle
\]

### Step 1: Compute the trigonometric values
- \(\cos\left(-\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}}\)
- \(\sin\left(-\frac{3\pi}{4}\right) = \frac{1}{\sqrt{2}}\)

### Step 2: Write the component form of the vector
Thus, the unit vector in component form is:

\[
\mathbf{v} = \langle -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \rangle
\]

Alternatively, you could express it as:

\[
\mathbf{v} = \langle -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \rangle
\]

This is the component form of the unit vector.

Would you like further details or have any other questions?

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Here are 5 related questions:
1. How do you convert angles from radians to degrees?
2. What is the general formula for finding the magnitude of any vector?
3. How do you find the direction of a vector if given in component form?
4. Can you explain how unit vectors are used in physics?
5. What is the significance of the negative angle in this problem?

**Tip:** Always remember that a unit vector has a magnitude of 1, which simplifies many vector calculations!

Find the component form of the unit vector that makes the angle -3pi/4 with the positive x-axis.

This solution explains how to find the component form of a unit vector that makes an angle of -3π/4 with the positive x-axis. Using trigonometric identities, the vector is computed step-by-step. Ideal for students in Grades 10-12.

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