Question

Find the vector equation of the line passing through \((9,1,2)\) and which is parallel to the vector \((1,1,1)\).

Short answer

Question: Find the vector equation of the line passing through the point (9, 1, 2) and parallel to the vector (1, 1, 1). Answer: The vector equation of the line is given by: \( \vec{r}(t) = \begin{pmatrix}9+t\\1+t\\2+t\end{pmatrix}\)

Step-by-step solution

Write down the given information

We are given the point \((9,1,2)\) through which the line passes and the vector parallel to the line, which is \((1,1,1)\). We need to find the vector equation of this line.

Write the position vector for the given point

The position vector of the given point \((9,1,2)\) is \(\begin{pmatrix}9\\1\\2\end{pmatrix}\).

Write the direction vector

The direction vector for the line, which is parallel to the given vector \((1,1,1)\), is simply the same vector, which in component form is: \(\begin{pmatrix}1\\1\\1\end{pmatrix}\).

Write the general vector equation for the line

The general vector equation for a line is given by the position vector of a point on the line plus a scalar multiple of the direction vector of the line. In this case, we have: \(\vec{r}(t) = \begin{pmatrix}9\\1\\2\end{pmatrix} + t\begin{pmatrix}1\\1\\1\end{pmatrix}\)

Write the final vector equation for the line

Expanding the above expression, we have: \( \vec{r}(t) = \begin{pmatrix}9+t\\1+t\\2+t\end{pmatrix}\) So, the vector equation of the line passing through \((9,1,2)\) and parallel to the given vector \((1,1,1)\) is: \( \vec{r}(t) = \begin{pmatrix}9+t\\1+t\\2+t\end{pmatrix}\)

Key concepts

Position Vector

Consider a line in 3D space; its position at any point can be represented by what’s known as a position vector. This vector originates from the origin of the coordinate system—think of it as the starting block in a race—and points directly to a specific location on the line. If we have a point on the line, say \( (9,1,2) \), the position vector for this point is visually an arrow pointing from the origin to that point, and mathematically it's written as \( \begin{pmatrix}9\1\2\end{pmatrix} \).

Understanding it in our context—the position vector essentially 'pins' the line in space and serves as a reference for all other points on the line to be described relative to this 'anchor'. It's the foundation upon which the vector equation of the line is built.

Direction Vector

Now let’s talk about the direction vector. Imagine hiking a trail that stretches as far as the eye can see in both directions. You only need to know one thing to walk the path: which way to go? That's the exact purpose of the direction vector in the context of a line. It tells you the orientation of the line or in which direction the line extends from the position you’ve already identified.

For our specific exercise, the direction vector is given as \( (1,1,1) \), indicating that the line moves equally along the x, y, and z axes. Regardless of where you're starting from on the line, if you follow the compass of this vector, you remain on the track.

Scalar Multiple

Now, there’s an element that allows us to explore any point along our line no matter how far it stretches—this is the scalar multiple. Think of it as a dial that you can turn up or down to 'slide' along the line. Mathematically, it's a simple number, denoted often by \( t \), that scales the direction vector. Dial \( t \)'s value up, and you move further away from your position vector; dial it down, and you move closer.

In the vector equation \( \vec{r}(t) = \begin{pmatrix}9+t\1+t\2+t\end{pmatrix} \), \( t \) is that scalar multiple. By changing its value, you're essentially travelling along the line to reach new points, as each value of \( t \) gives you a different point on the line.

Vector Components

The concept of vector components provides a method to effectively dismantle vectors into more digestible parts. It’s similar to having the ingredients listed out separately for a complex recipe. For every vector in three-dimensional space, we look at the components along the three axes—x, y, and z.

Our direction vector \( (1,1,1) \) is a great example. Its components are all '1', meaning it has equal parts along each axis. Understanding components is crucial, especially when you start adding vectors together, like when forming the vector equation of a line. Each component must be treated individually, quietly respecting the geometry of space, to ensure the equation accurately represents the line's true position and direction.