Question

The United States Capitol is

Answer 1

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 $\vec C_{1} =(-1,7)$ and $\vec C_{2} = (-7,13)$.

Step-by-step explanation:

At first we need to calculate the vector distance between $A(x,y) = (2, 4)$ and $B(x,y) =(-10,16)$ by following vectorial subtraction:

$\overrightarrow{AB} = \vec B - \vec A$ (Eq. 1)

Where:

$\overrightarrow{AB}$ - Vector distance between points A and B, dimensionless.

$\vec A$, $\vec B$ - Vector distance between each point and origin, dimensionless.

If we know that $A(x,y) = (2, 4)$ and $B(x,y) =(-10,16)$, then we have the following result:

$\overrightarrow {AB} = (-10,16)-(2,4)$

$\overrightarrow{AB} = (-10-2,16-4)$

$\overrightarrow{AB} = (-12,12)$

Besides, we can find the location of any point inside the line segment by using the following vectorial equation:

$\vec C = \vec A + r\cdot \overrightarrow{AB}$ (Eq. 2)

Where:

$r$ - Segment factor, dimensionless.

$\vec C$ - Location of resulting point, dimensionless.

There are two different options for the location of resulting point: $r_{1} = \frac{1}{4}$ and $r_{2} = \frac{3}{4}$. Now we proceed to find each option:

$r_{1} = \frac{1}{4}$

$\vec C_{1} = (2,4) +\frac{1}{4}\cdot (-12,12)$

$\vec C_{1} = (2,4)+(-3,3)$

$\vec C_{1} =(-1,7)$

$r_{2} = \frac{3}{4}$

$\vec C_{2} = (2,4) +\frac{3}{4}\cdot (-12,12)$

$\vec C_{2} = (2,4) +(-9,9)$

$\vec C_{2} = (-7,13)$

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 $\vec C_{1} =(-1,7)$ and $\vec C_{2} = (-7,13)$.