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The slope of a line is a measure of its steepness or incline. In 2D space, the slope is typically denoted by the symbol \( m \), and it describes how much the line rises or falls as it moves along the x-axis. In general terms, the slope is given by the change in the y-coordinate (vertical change) divided by the change in the x-coordinate (horizontal change). This can be expressed as:\[ m = \frac{\Delta y}{\Delta x} \]In the context of vector calculus, we often derive the slope from a vector equation describing the line. For example, given the vector equation \( \mathbf{r} = (1-2t)\mathbf{i} - (2-3t)\mathbf{j} \), the direction vector \( \mathbf{d} = \langle -2, -3 \rangle \) indicates the direction of the line.

• Change in y: The coefficient of \( \mathbf{j} \) is \(-3\).
• Change in x: The coefficient of \( \mathbf{i} \) is \(-2\).
• Slope \( m = \frac{-3}{-2} = \frac{3}{2} \).

This slope of \( \frac{3}{2} \) tells us that for a horizontal movement of 2 units in the positive x-direction, the line rises by 3 units in the y-direction, indicating an upward, rightward incline.

A vector equation is a mathematical way to describe a line or curve in space using vectors. It allows us to track the position of points along the line as a function of a parameter, typically denoted by \( t \). The general form of a vector equation for a line can be written as \( \mathbf{r} = \mathbf{a} + t \mathbf{d} \), where:

• \( \mathbf{a} \) is a position vector to a specific known point on the line.
• \( \mathbf{d} \) is the direction vector for the line, giving its orientation.
• \( t \) is a scalar parameter that signifies how far and in what direction to move along the line from \( \mathbf{a} \).

For example, in the vector equation \( \mathbf{r} = (2+t) \mathbf{i} + (1-2t) \mathbf{j} + 3t \mathbf{k} \),

• \( \mathbf{a} = \langle 2, 1, 0 \rangle \), is an initial point on the line.
• \( \mathbf{d} = \langle 1, -2, 3 \rangle \) represents the direction in which the line extends.

By varying \( t \), we can find different points on the line, thus fully describing its path in space. This form is particularly useful in vector calculus for solving geometric problems like intersection and projections.

The intersection of planes refers to finding a point or line where two planes meet. In a 3D space context, determining where a line intersects another plane is key to solving many geometric and algebraic problems. Intersection problems typically involve finding the value of the parameter \( t \) that satisfies a certain condition.For instance, consider the line defined by the equation \( \mathbf{r} = (2+t) \mathbf{i} + (1-2t) \mathbf{j} + 3t \mathbf{k} \) intersecting the \( xz \)-plane.

• The \( xz \)-plane is characterized by its \( y \)-coordinate being zero (i.e., \( y = 0 \)).
• The condition for intersection is setting the \( y \)-component of the vector equation to zero: \( 1 - 2t = 0 \).

Solving this gives \( t = \frac{1}{2} \). Substituting back, we find:

• \( x = 2 + \frac{1}{2} = \frac{5}{2} \)
• \( y = 0 \)
• \( z = 3 \cdot \frac{1}{2} = \frac{3}{2} \)

Thus, the intersection point at \( \frac{5}{2}, 0, \frac{3}{2} \) shows where the line pierces the \( xz \)-plane. Understanding this process is crucial in vector calculus for visualizing spatial relationships and solving complex problems.