Convert v=8(cos135degressi+sin135degressj)

To convert the vector \( \mathbf{v} = 8(\cos 135^\circ \mathbf{i} + \sin 135^\circ \mathbf{j}) \) into its rectangular (or Cartesian) form, we need to calculate the values of \( \cos 135^\circ \) and \( \sin 135^\circ \).

### Step 1: Calculate the trigonometric values
- \( \cos 135^\circ = -\frac{1}{\sqrt{2}} \)
- \( \sin 135^\circ = \frac{1}{\sqrt{2}} \)

### Step 2: Substitute these values into the vector equation
\[
\mathbf{v} = 8\left(-\frac{1}{\sqrt{2}} \mathbf{i} + \frac{1}{\sqrt{2}} \mathbf{j}\right)
\]

### Step 3: Multiply by the scalar \( 8 \)
\[
\mathbf{v} = -8\frac{1}{\sqrt{2}} \mathbf{i} + 8\frac{1}{\sqrt{2}} \mathbf{j}
\]

### Step 4: Simplify the expressions
Since \( \frac{1}{\sqrt{2}} \) is the same as \( \frac{\sqrt{2}}{2} \):
\[
\mathbf{v} = -8 \times \frac{\sqrt{2}}{2} \mathbf{i} + 8 \times \frac{\sqrt{2}}{2} \mathbf{j}
\]

\[
\mathbf{v} = -4\sqrt{2} \mathbf{i} + 4\sqrt{2} \mathbf{j}
\]

### Final Answer
The vector in rectangular form is:
\[
\mathbf{v} = -4\sqrt{2} \mathbf{i} + 4\sqrt{2} \mathbf{j}
\]

Would you like more details on this process, or do you have any questions?

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**Relative Questions:**
1. How do you calculate the cosine and sine of angles in different quadrants?
2. What are some applications of converting vectors from polar to rectangular form?
3. How does the magnitude of the vector relate to its components in rectangular form?
4. Can you explain why \( \cos 135^\circ \) is negative and \( \sin 135^\circ \) is positive?
5. How would this vector look graphically in the coordinate plane?

**Tip:** Always remember that angles measured in degrees can be converted to radians when working with trigonometric functions, especially in calculus.

Learn how to convert the vector v=8(cos135degressi+sin135degressj) into its rectangular form using trigonometric functions. Understand the step-by-step process involving cosine and sine values of angles. Suitable for advanced high school students and beyond.

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