Determine the equation of the line through the origin and the point P(10,3)

a) r=s(10,3), seR

b) x=3t, y=10t, teR

c) r=s(-3,10), seR

d) r=s(3,10), seR

suppose I wrote it this way:

r = (0,0) + s(10,3)

what do you think?

To determine the equation of the line through the origin and the point P(10,3), we can use the slope-intercept form of the equation of a line, which is y = mx + b.

The slope (m) can be found using the coordinates of the two points. The slope is given by:

m = (y2 - y1) / (x2 - x1)

In this case:
x1 = 0 (origin x-coordinate)
y1 = 0 (origin y-coordinate)
x2 = 10 (P x-coordinate)
y2 = 3 (P y-coordinate)

Substituting these values into the formula, we get:

m = (3 - 0) / (10 - 0)
m = 3/10

So, the slope of the line is 3/10.

Since the line passes through the origin (0,0), the y-intercept (b) is also 0.

Therefore, the equation of the line through the origin and the point P(10,3) is:

y = (3/10)x + 0

Simplifying the equation, we have:

y = (3/10)x

Therefore, the correct answer is option b) x=3t, y=10t, t∈R.

To determine the equation of the line through the origin and the point P(10,3), we first need to calculate the slope of the line.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula m = (y2 - y1) / (x2 - x1).

In this case, the two points are the origin (0,0) and P(10,3).

Using the formula, the slope is: m = (3 - 0) / (10 - 0) = 3/10.

Now let's look at the answer choices:

a) r = s(10,3), seR
This answer choice does not describe the equation of a line. It is simply a vector representation.

b) x = 3t, y = 10t, teR
This answer choice represents a line, but it does not pass through the origin (0,0) as required.

c) r = s(-3,10), seR
This answer choice also does not pass through the origin. The point (-3, 10) is not the origin.

d) r = s(3,10), seR
This answer choice represents the equation of the line passing through the origin (0,0) and the point P(10,3).

Therefore, the correct answer is d) r = s(3,10), seR.