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!