Answer:
The two different point on a segment joining the United States Capital and the White House such that the ratio of the shorter segments created by each is 1 : 3 are [tex]\vec C_{1} =(-1,7)[/tex] and [tex]\vec C_{2} = (-7,13)[/tex].
Step-by-step explanation:
At first we need to calculate the vector distance between [tex]A(x,y) = (2, 4)[/tex] and [tex]B(x,y) =(-10,16)[/tex] by following vectorial subtraction:
[tex]\overrightarrow{AB} = \vec B - \vec A[/tex] (Eq. 1)
Where:
[tex]\overrightarrow{AB}[/tex] - Vector distance between points A and B, dimensionless.
[tex]\vec A[/tex], [tex]\vec B[/tex] - Vector distance between each point and origin, dimensionless.
If we know that [tex]A(x,y) = (2, 4)[/tex] and [tex]B(x,y) =(-10,16)[/tex], then we have the following result:
[tex]\overrightarrow {AB} = (-10,16)-(2,4)[/tex]
[tex]\overrightarrow{AB} = (-10-2,16-4)[/tex]
[tex]\overrightarrow{AB} = (-12,12)[/tex]
Besides, we can find the location of any point inside the line segment by using the following vectorial equation:
[tex]\vec C = \vec A + r\cdot \overrightarrow{AB}[/tex] (Eq. 2)
Where:
[tex]r[/tex] - Segment factor, dimensionless.
[tex]\vec C[/tex] - Location of resulting point, dimensionless.
There are two different options for the location of resulting point: [tex]r_{1} = \frac{1}{4}[/tex] and [tex]r_{2} = \frac{3}{4}[/tex]. Now we proceed to find each option:
[tex]r_{1} = \frac{1}{4}[/tex]
[tex]\vec C_{1} = (2,4) +\frac{1}{4}\cdot (-12,12)[/tex]
[tex]\vec C_{1} = (2,4)+(-3,3)[/tex]
[tex]\vec C_{1} =(-1,7)[/tex]
[tex]r_{2} = \frac{3}{4}[/tex]
[tex]\vec C_{2} = (2,4) +\frac{3}{4}\cdot (-12,12)[/tex]
[tex]\vec C_{2} = (2,4) +(-9,9)[/tex]
[tex]\vec C_{2} = (-7,13)[/tex]
The two different point on a segment joining the United States Capital and the White House such that the ratio of the shorter segments created by each is 1 : 3 are [tex]\vec C_{1} =(-1,7)[/tex] and [tex]\vec C_{2} = (-7,13)[/tex].
