Convert v=8(cos135degressi+sin135degressj)to liner form

To express the vector \( \mathbf{v} = 8(\cos 135^\circ \mathbf{i} + \sin 135^\circ \mathbf{j}) \) in linear equation form, we interpret the vector as describing a line in two dimensions. This vector points from the origin along the direction given by the angle \( 135^\circ \) with a magnitude of 8.

### Step 1: Find the unit vector direction
First, we calculate the unit vector in the direction of \( 135^\circ \).

\[
\hat{\mathbf{v}} = \cos 135^\circ \mathbf{i} + \sin 135^\circ \mathbf{j}
\]
\[
\hat{\mathbf{v}} = -\frac{1}{\sqrt{2}} \mathbf{i} + \frac{1}{\sqrt{2}} \mathbf{j}
\]

### Step 2: Express the vector in linear form
The linear form of the vector can be written as:
\[
\mathbf{v} = x \mathbf{i} + y \mathbf{j}
\]
where \( x = -4\sqrt{2} \) and \( y = 4\sqrt{2} \), derived from the rectangular components.

The vector equation \( \mathbf{v} = t(-4\sqrt{2} \mathbf{i} + 4\sqrt{2} \mathbf{j}) \) for a scalar \( t \) represents a line, which can be expressed in parametric form:
\[
x = -4\sqrt{2} t
\]
\[
y = 4\sqrt{2} t
\]

### Step 3: Eliminate the parameter \( t \)
To find the linear equation, eliminate \( t \):
\[
t = \frac{x}{-4\sqrt{2}} = \frac{y}{4\sqrt{2}}
\]
\[
\frac{x}{-4\sqrt{2}} = \frac{y}{4\sqrt{2}}
\]

### Step 4: Simplify the equation
By cross-multiplying:
\[
x = -y
\]

### Final Answer
The vector \( \mathbf{v} = 8(\cos 135^\circ \mathbf{i} + \sin 135^\circ \mathbf{j}) \) represents the line with the equation:
\[
x + y = 0
\]

This is the linear form of the vector, indicating a line passing through the origin with a slope of -1.

Do you want more details or have any questions?

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**Relative Questions:**
1. How does the slope of the line relate to the direction of the vector?
2. How do you find the line equation from a vector in three dimensions?
3. What is the significance of the line passing through the origin in this context?
4. How would you express this line equation in different forms, such as slope-intercept?
5. How does changing the magnitude of the vector affect the linear equation?

**Tip:** Always ensure that when converting vectors to line equations, you consider the direction and magnitude to get the correct form.

Learn how to convert the vector v = 8(cos 135°i + sin 135°j) into linear equation form. Understand the process step-by-step with detailed explanations and calculations, including finding unit vector direction, expressing in linear form, and deriving the line equation. Suitable for high school students and anyone interested in vector mathematics.

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