Final answer:
The component form of the vector is <3, 2>, the magnitude is approximately 3.6 units, and the direction is approximately 33.7° from the positive x-axis.
Step-by-step explanation:
To find the component form of a vector with an initial point at (2, 5) and a terminal point at (5, 7), we subtract the coordinates of the initial point from those of the terminal point:
- x-component: 5 - 2 = 3
- y-component: 7 - 5 = 2
The component form of the vector is therefore <3, 2>.
To calculate the magnitude of the vector, we use the Pythagorean theorem:
Magnitude = √(x-component² + y-component²)
= √(3² + 2²)
= √(9 + 4)
= √13
≈ 3.6 (to the nearest tenth of a unit).
To determine the direction of the vector, we find the angle it makes with the positive x-axis. This can be done using the arctangent function:
Direction angle = arctan(y-component/x-component)
= arctan(2/3)
≈ 33.7° (to the nearest tenth of a degree).