Final answer:
To determine how far and in what direction Alisha must travel from her cousin's restaurant to her grandma's house, we can construct a right-angle triangle with the given vectors and use trigonometric laws to calculate the remaining side and angle.
Step-by-step explanation:
Alisha's journey can be visualized using vector addition, where the vector from her house to her cousin's restaurant and the vector from her cousin's restaurant to her grandma's house must add up to the direct distance between her house and her grandma's house, which is 345 km due North. Since the journey to the cousin's restaurant goes 135 km at a direction of N 50° E, we'll need to calculate the remaining vector to Alisha's grandma's house.
We know that Alisha's cousin's restaurant is thus to the North-East of her starting position. To find the displacement from her cousin's restaurant to her grandma's house, imagine a right-angle triangle where the hypotenuse is the path from her house to her grandma's house, one leg is the path to her cousin's restaurant, and the other leg is the unknown path from the cousin's restaurant to grandma's house.
The angle at Alisha's starting point is 50°. To find the angle at the cousin's restaurant, we subtract this from 90° since her grandma's house is due North of her starting position, which results in a 40° angle (90° - 50°). Using the cosine law and sine law, we can then calculate the length of the remaining side (distance to grandma's house) and the angle of travel from the cousin's restaurant to grandma's house.