Given the vectors AB = (7, -2) and BC = (-3, 5), work out how much further Pru travels than Yang when Pru travels from A to B and then B to C, and Yang travels directly from A to C... Given the vectors AB = (7, -2) and BC = (-3, 5), work out how much further Pru travels than Yang when Pru travels from A to B and then B to C, and Yang travels directly from A to C. Provide your answer in km, correct to one decimal place. Read more

Steps to Solve

1. Calculate the distance Pru travels (AB + BC)

Pru travels from A to B and then from B to C. To find the total distance, we first need to calculate the individual distances.

For segment $AB$, represented as vector $\overrightarrow{AB} = (7, -2)$:

• The distance from $A$ to $B$ can be found using the formula for the magnitude of a vector: $$ |AB| = \sqrt{(7)^2 + (-2)^2} $$

Calculating this gives: $$ |AB| = \sqrt{49 + 4} = \sqrt{53} \approx 7.28 \text{ km} $$

Now for segment $BC$, represented as vector $\overrightarrow{BC} = (-3, 5)$:

• The distance from $B$ to $C$ is calculated similarly: $$ |BC| = \sqrt{(-3)^2 + (5)^2} $$

Calculating this gives: $$ |BC| = \sqrt{9 + 25} = \sqrt{34} \approx 5.83 \text{ km} $$

Now, adding both distances for Pru's total distance: $$ \text{Total distance (Pru)} = |AB| + |BC| = \sqrt{53} + \sqrt{34} \approx 7.28 + 5.83 \approx 13.11 \text{ km} $$

2. Calculate the distance Yang travels (AC)

Yang travels directly from A to C. We need to determine the vector $\overrightarrow{AC}$ which can be found by adding vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$: $$ \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = (7, -2) + (-3, 5) = (7 - 3, -2 + 5) = (4, 3) $$

Next, we calculate the distance from $A$ to $C$: $$ |AC| = \sqrt{(4)^2 + (3)^2} $$

Calculating gives: $$ |AC| = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ km} $$

3. Calculate the difference in distance traveled

Now, we can find out how much further Pru travels compared to Yang: $$ \text{Difference} = \text{Total distance (Pru)} - |AC| $$

Substituting the values gives: $$ \text{Difference} = 13.11 - 5 = 8.11 \text{ km} $$

Thus, rounding the answer to one decimal place: $$ \text{Difference} \approx 8.1 \text{ km} $$