Let A = (−6, −4), B = (3, 2), and C = (6, 4).
(a)These points lie on a line through the origin. Find its slope.
(b)Let u be the vector whose components are the x-coordinates of A, B, and C, and let v
be the vector whose components are the y-coordinates of A, B, and C. Show that v is a
positive scalar multiple of u (thus u and v point in the same direction).
(c)Explain why the scalar multiple in part (b) equals the slope you found in part (a).
(d)What would the vectors u and v have looked like if A, B, and C had not been collinear
with the origin?
slope of AB is (2+4)/(3+6) = 2/3
since the slope is 2/3, v = 2/3 u
if the points started at (h,k) then there'd be an offset of
u = u+h
v = v+k
(a) To find the slope of the line passing through the origin and the points A, B, and C, we can use the formula for slope:
m = (y2 - y1) / (x2 - x1)
For the origin O=(0,0) and point A=(-6, -4):
m1 = (-4 - 0) / (-6 - 0)
= -4 / -6
= 2/3
Similarly, for the points B=(3, 2) and C=(6, 4):
m2 = (2 - 0) / (3 - 0)
= 2 / 3
m3 = (4 - 0) / (6 - 0)
= 4 / 6
= 2 / 3
Since all three slopes are equal to 2/3, the points A, B, and C lie on a line through the origin with a slope of 2/3.
(b) Let's calculate the vectors u and v:
u = [x1, x2, x3]
= [-6, 3, 6]
v = [y1, y2, y3]
= [-4, 2, 4]
We can see that v is a scalar multiple of u if we can find a scalar c such that v = c * u. Let's calculate c:
c = v1 / u1
= -4 / -6
= 2/3
Thus, v is a positive scalar multiple of u.
(c) The scalar multiple in part (b) equals the slope found in part (a) because the scalar multiple c is equal to the slope m. In this case, c = m = 2/3.
(d) If A, B, and C were not collinear with the origin, the vectors u and v would have different components. For example, if the points were A=(1, 2), B=(3, 4), and C=(5, 6), then:
u = [1, 3, 5]
v = [2, 4, 6]
In this case, v would not be a scalar multiple of u because the y-components are not proportional to the x-components.
(a) To find the slope of a line passing through two points, we use the formula:
slope = (change in y)/(change in x)
Let's calculate the slope using points A and B:
slope = (2 - (-4))/(3 - (-6))
= 6/9
= 2/3
Therefore, the slope of the line passing through the origin and points A and B is 2/3.
(b) To show that vector v is a positive scalar multiple of u, we need to compare their respective components.
u = [x-coordinates of A, B, and C] = [-6, 3, 6]
v = [y-coordinates of A, B, and C] = [-4, 2, 4]
To see if v is a scalar multiple of u, we check if each corresponding component of v is a positive multiple of the corresponding component of u:
For the first component:
-4/(-6) = 2/3
For the second component:
2/3 = 2/3
For the third component:
4/6 = 2/3
As we can see, each component of v is a positive multiple (specifically 2/3) of the corresponding component of u. Therefore, v is a positive scalar multiple of u.
(c) The scalar multiple in part (b) equals the slope found in part (a) because the components of v are scalar multiples of the components of u. In other words, the ratio of the y-components to the x-components of points A, B, and C is the same as the slope.
(d) If points A, B, and C were not collinear with the origin, the vectors u and v would not have the same characteristics. Specifically, u would no longer represent the x-coordinates and v would no longer represent the y-coordinates. The vectors u and v would instead represent the displacement between the points A, B, and C and the origin. The scalar multiple relationship between u and v would not hold, and the slope of the line passing through the origin and points A, B, and C would also be different.