Final answer:
The number of routes from (0,0) to (4,7) can be calculated as combinations C(11, 4). Displacement is the same regardless of the path if the start and end points are the same. A 2-D vector is the sum, not the product, of its x and y components, and it can form a right-angle triangle with these components.
Step-by-step explanation:
To find the number of routes that start at point (0,0) and end at (4,7) on an XY coordinate plane, you need to calculate the number of combinations for the moves. Since it's only possible to move one unit to the right or one unit up, and a route to (4,7) requires 4 steps right and 7 steps up, a total of 11 steps are required. The number of different routes is equivalent to the number of ways to choose 4 right moves out of 11 moves, which is a combinations problem and can be solved using the formula for combinations: C(n, r) = n! / (r!(n-r)!). In this case, it's C(11, 4).
Vector Displacement
For the true/false statement about displacement, the answer is false. Whether a person walks 2 blocks east and 5 blocks north or 5 blocks north and then 2 blocks east, the displacement will be the same because displacement is a vector quantity that depends only on the initial and final positions, not on the path taken. Both paths describe a right-angled triangle with legs of 2 and 5 blocks, resulting in the same hypotenuse.
Vectors and Components
It's false that every 2-D vector can be expressed as the product of its x and y-components. Instead, a 2-D vector is represented as the sum of its x and y-components, often written in component form (Ax, Ay) or as a magnitude and direction.
The slope for a line passing through points (1, 0.1) and (7, 26.8) can be calculated using the slope formula (rise over run), which gives us 26.8 - 0.1 / 7 - 1 = 26.7 / 6 = 4.45.
The statement that a vector can form the shape of a right-angle triangle with its x and y components is true. This is because those components are perpendicular to each other, representing the legs of a right-angle triangle with the vector itself being the hypotenuse.