Question

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(-2,5)\), parallel to \(\vec{d}=\langle 0,1\rangle\).

Short answer

Vector Equation: \(\vec{r} = \langle -2, 5 \rangle + t\langle 0, 1 \rangle\); Parametric: \(x = -2\), \(y = 5 + t\); Symmetric: \(x = -2\), \(y = 5 + t\).

Step-by-step solution

Identify Given Information

The line passes through point \(P = (-2, 5)\) and is parallel to vector \(\vec{d} = \langle 0, 1 \rangle\).

Write the Vector Equation of the Line

The vector equation of a line can be expressed as \(\vec{r} = \vec{r_0} + t\vec{d}\), where \(\vec{r_0}\) is a position vector of a point on the line, and \(\vec{d}\) is the direction vector. Here, \(\vec{r_0} = \langle -2, 5 \rangle\) and \(\vec{d} = \langle 0, 1 \rangle\). Thus, the vector equation is \(\vec{r} = \langle -2, 5 \rangle + t\langle 0, 1 \rangle\).

Convert Vector Equation to Parametric Equations

From the vector equation \(\vec{r} = \langle -2, 5 \rangle + t\langle 0, 1 \rangle\), we split into components: \(x = -2 + 0t = -2\) and \(y = 5 + 1t = 5 + t\). Therefore, the parametric equations are \(x = -2\) and \(y = 5 + t\).

Write the Symmetric Equation of the Line

The symmetric form of a line is \(\frac{x - x_0}{a} = \frac{y - y_0}{b}\) where \(\langle a, b \rangle\) is the direction vector. Using point \((-2, 5)\) and direction vector \(\langle 0, 1 \rangle\), we obtain \(\frac{x + 2}{0} = \frac{y - 5}{1}\). As division by zero isn't defined for the \(x\) component, the symmetric form reflects \(x = -2\) with \(y\) unrestricted by \(\frac{y - 5}{1} = 0\) or \(y = 5 + t\).

Key concepts

Parametric Equations

Parametric equations are a powerful way to describe a line in space by using a parameter, usually denoted as \(t\). In our scenario, we have a line that passes through a specific point and runs parallel to a direction vector. The line can be expressed as an equation based on this information.

Let's break down the concept into digestible parts:

• The standard parametric form is expressed as two separate equations: one for \(x\) and the other for \(y\), based on time \(t\).
• These equations link the parameter \(t\) with the line's coordinates. For the given line, the parametric equations are \(x = -2\) and \(y = 5 + t\).
• Here, \(x = -2\) remains constant as the direction vector affects only the \(y\) variable with changes in parameter \(t\).

This representation is invaluable because it allows us to express complex paths simply by altering \(t\). It provides insight into where the line and its components (\(x\) and \(y\)) lie at any chosen parameter value.

Symmetric Equations

Symmetric equations are another elegant way to represent lines, particularly useful when handling two or more variables. This form gives us symmetrical expressions for each coordinate, based directly on known line properties.

To understand symmetric equations better:

• The general form uses the line's direction vector, \(\langle a, b \rangle\), and a point on the line, \((x_0, y_0)\).
• The equation is often written as \(\frac{x - x_0}{a} = \frac{y - y_0}{b}\).
• For our line, the symmetric form collapses to a simpler form since the \(x\) direction vector component is zero: \(x = -2\).
• You'll notice one part of the equation is omitted (\(\frac{x+2}{0}\)) due to the undefined nature of dividing by zero, emphasizing the fixed \(x\) coordinate.

This form outlines that for every unit increase in \(t\), \(y\) increases while \(x\) remains steady. This provides us with a cleaner way to comprehend the spatial alignment and affinity of this particular line.

Direction Vector

The direction vector is a fundamental concept in vector equations, lending directionality to how we express lines in geometric settings. In our case, this vector is crucial as it defines how the line moves through space.

Here's what you need to know:

• The direction vector \(\vec{d}\), in this example, is \(\langle 0, 1 \rangle\). This tells us that any point on the line moves along the \(y\)-axis.
• The vector components \(\langle a, b \rangle\) serve as multipliers for how far and in which direction to move: 0 in the \(x\) direction (no movement), and 1 in the \(y\) direction (movement).
• Therefore, the vector establishes that our line is vertical with fixed \(x\) values, but \(y\) changes with parameter \(t\).

The direction vector's role is vital; it dictates the line's trajectory, demonstrating how variations in \(t\) manifest spatially. Understanding this helps us grasp the bigger picture of vector space relationships.