Question

Write down the vector equation of the line passing through the points with position vectors $$ \begin{aligned} &p=3 i+7 j-2 k \\ &q=-3 i+2 j+2 k \end{aligned} $$ Find also the cartesian equation of this line.

Short answer

Question: Find the vector and cartesian equations of the line passing through the points with position vectors p = 3i + 7j - 2k and q = -3i + 2j + 2k. Answer: Vector Equation: $$ \vec{r} = (3 - 6t)i + (7 - 5t)j + (-2 + 4t)k $$ Cartesian Equation: $$ \frac{x - 3}{-6} = \frac{y - 7}{-5} = \frac{z + 2}{4} $$

Step-by-step solution

Find the direction vector of the line

The direction vector of the line passing through the points with position vectors p and q can be found by subtracting the position vector p from q. So, the direction vector, d, is given by: $$ d = q - p $$ Now, substitute the given position vectors and find the direction vector: $$ \begin{aligned} d &= (-3i + 2j + 2k) - (3i + 7j - 2k) \\ &= -3i + 2j + 2k - 3i - 7j + 2k \\ &= (-3 - 3)i + (2 - 7)j + (2 + 2)k \\ &= -6i - 5j + 4k \end{aligned} $$ So, the direction vector d is: \(-6i - 5j + 4k\)

Write the vector equation of the line

The vector equation of the line passing through a point with position vector p and having a direction vector d is given by: $$ \vec{r} = p + td $$ Now, substitute the given position vector p and the direction vector d found in step 1: $$ \vec{r} = (3i + 7j - 2k) + t(-6i - 5j + 4k) $$ So, the vector equation of the line is: \(\vec{r} = (3 - 6t)i + (7 - 5t)j + (-2 + 4t)k\)

Convert the vector equation to cartesian equation

To find the cartesian equation of the line, first express i, j, and k components of the vector equation in terms of x, y, and z: $$ \begin{cases} x = 3 - 6t \\ y = 7 - 5t \\ z = -2 + 4t \end{cases} $$ Now, solve the first equation for variable t: $$ t = \frac{x - 3}{-6} $$ Do the same for the second and third equation: $$ \begin{cases} t = \frac{y - 7}{-5} \\ t = \frac{z + 2}{4} \end{cases} $$ Now, equate all expressions for t: $$ \frac{x - 3}{-6} = \frac{y - 7}{-5} = \frac{z + 2}{4} $$ So, the cartesian equation of the line is: \(\frac{x - 3}{-6} = \frac{y - 7}{-5} = \frac{z + 2}{4}\)

Key concepts

Cartesian Equation of a Line

Understanding the Cartesian equation of a line is crucial for visualizing and solving geometrical problems related to lines in a two- or three-dimensional space. The Cartesian equation represents a line using linear equations that relate the coordinates of any point on the line.

For instance, considering the vector equation of the line \( \vec{r} = (3 - 6t)i + (7 - 5t)j + (-2 + 4t)k \), we can find a Cartesian equation by isolating the parameter 't' and setting the resulting expressions equal to each other. This yields \( \frac{x - 3}{-6} = \frac{y - 7}{-5} = \frac{z + 2}{4} \), which succinctly expresses that the ratio of the differences in coordinates from a certain point on the line and their respective directional changes are equal for all points on the line.

• Provides a straightforward way to verify if a given point lies on the line
• Used for finding the intersection point of two lines or a line and a plane
• Essential in understanding the geometry of lines in higher dimensions

Position Vectors

Position vectors serve as an anchor, allowing us to represent the location of any point in space with respect to a fixed point, usually the origin of the coordinate system. In our example, we are given the position vectors of two points \( p = 3 i + 7 j - 2 k \) and \( q = -3 i + 2 j + 2 k \).

Crucial Role of Position Vectors

Position vectors are foundational when using vector methods to solve geometrical problems:

• They allow the translation of geometric points into an algebraic form
• Act as starting points to explore properties like distance, angles, and more
• Essential in defining vectors like the direction vector, which is derived from two position vectors

It is the subtraction of the position vector \( p \) from \( q \) that gives us the direction vector needed to formulate the equation of the line, illustrating the interconnectedness of these vector notions.

Direction Vector

The direction vector is pivotal as it indicates the orientation and 'steer' of the line in three-dimensional space. For the line in our exercise, the direction vector is calculated as \( d = q - p \) which simplifies to \( -6i - 5j + 4k \).

This vector not only points in the direction of the line but also determines the slope of the line in each dimension:

• It is the vector parallel to the line and passes through the two given points
• Reveals how rapidly the line ascends or descends along each axis
• Used in defining the vector equation of a line

In essence, it is the difference between the position vectors that gives us this crucial piece of the puzzle, the direction vector, which is then used to express the infinite collection of points that form the line in space.