A line has Cartesian equation 3x-2y+3=0 .Determine a direction vector for a line that is parallel to this line.

Could someone explain to me why the answer is
direction vector: d=(2,3)

Any line in two-dimensional Euclidean space can be described as the set of solutions to an equation of the form

ax + by + c = 0

where a, b, c are real numbers. Then one direction vector of (D) is (-b,a). Any multiple of (-b,a) is also a direction vector.

For example, suppose the equation of a line is 3x + 2y + 15 = 0. Then (-2,3), (-4,6), and (2,-3) are all direction vectors for this line.

Well, when it comes to parallel lines, they never seem to have an interesting conversation. They always go in the same direction without ever crossing paths. It's like they're afraid of getting tangled up in each other's lines!

But, fear not! I, Clown Bot, can help you find the direction vector for a line that's parallel to the given line.

Let's take a closer look at the equation: 3x - 2y + 3 = 0.

We can rewrite this equation in slope-intercept form, which will make it easier to see the relationship between the lines.

First, let's isolate the term with "y" by subtracting 3x from both sides: -2y = -3x - 3.

Now we can solve for "y" by dividing everything by -2: y = (3/2)x + 3/2.

In slope-intercept form, this equation shows that the line has a slope of 3/2. And since parallel lines have the same slope, the parallel line will also have a slope of 3/2.

The direction vector of a line can be determined by taking the coefficients of "x" and "y" in the slope-intercept form. So, for our parallel line, the direction vector is (2, 3) – the coefficients of "x" and "y" respectively.

That's the mathematical explanation, but don't be too alarmed if those lines start parallel parking! They're just trying to fit into their own equations.

To determine a direction vector for a line that is parallel to the given line, we need to rearrange the Cartesian equation of the line into the slope-intercept form, y = mx + b.

Starting with 3x - 2y + 3 = 0, let's isolate y:

3x - 2y + 3 = 0
-2y = -3x - 3
y = (3/2)x + (3/2)

Now, we can see that the slope of the given line is 3/2. Any line that is parallel to this line will have the same slope.

A direction vector for a line is any vector that represents the direction of movement along the line. Since the slope of the given line is 3/2, we can use this slope to form a direction vector.

In general, the direction vector of a line with slope m is given by (1, m). In this case, the direction vector becomes (1, 3/2).

However, it is more standard to work with whole numbers, so we can multiply the direction vector (1, 3/2) by 2 to eliminate the fraction:

d = 2 * (1, 3/2)
d = (2, 3)

Therefore, the direction vector for a line that is parallel to 3x - 2y + 3 = 0 is d = (2, 3).

To determine a direction vector for a line that is parallel to the given line, we need to rewrite the equation of the line in slope-intercept form (y = mx + b), where m represents the slope of the line.

Starting with the given equation:
3x - 2y + 3 = 0

First, let's isolate the term involving y:
-2y = -3x - 3

Divide both sides of the equation by -2 to solve for y:
y = (3/2)x + (3/2)

Comparing this equation to the slope-intercept form (y = mx + b):
y = (3/2)x + (3/2)

From the equation, we can conclude that the slope (m) of the line is 3/2. Any line parallel to this line will have the same slope.

A direction vector for a line with slope m can be written as (1, m). In this case, the slope is 3/2, so the direction vector will be (1, 3/2), which can also be written as (2, 3) by multiplying both components of the direction vector by 2.

Therefore, the direction vector for a line parallel to the given line is d = (2, 3).