Final answer:
Several true/false statements about vectors have been explored. It is true that the Pythagorean theorem can be applied to find the resultant vector of two perpendicular vectors. However, angles alone are insufficient to determine the resultant vector's direction, and a person's displacement remains the same if the order of their movements is commuted.
Step-by-step explanation:
Regarding the perception of vectors in mathematics, several true/false statements have been presented, questioning the application of vector addition and the characteristics of vectors. Let's address each one:
- Pythagorean Theorem: It is true that we can use the Pythagorean theorem to calculate the length of the resultant vector obtained from the addition of two vectors which are at right angles to each other. This is because when vectors are at right angles, they form a right-angled triangle, and the resultant vector corresponds to the hypotenuse of that triangle.
- Resultant Angle: The statement is false. If only the angles of two vectors are known, without knowledge of their magnitudes, we cannot determine the angle of their resultant addition vector. Angles alone are insufficient; magnitude information is also needed.
- Right Angle Triangle: A vector can indeed form the shape of a right angle triangle with its x and y components. This is true. The x and y components act as the perpendicular sides of the triangle, while the vector itself is the hypotenuse.
- Equal Displacement: This statement is false. A person walking 2 blocks east and 5 blocks north will have the same displacement as another person walking 5 blocks north and then 2 blocks east as these are commutative actions and will result in the same diagonal in a right angled triangle.
- Magnitude and Direction: It is true that with the angles of two vectors and the magnitude of one, we can find the magnitude and direction of the resultant vector. This can be done using trigonometric methods and vector addition principles.
The distinctions made here showcase various properties of vectors and how we handle vector addition in mathematics. Each statement provides insight into the geometric and algebraic methods employed to solve problems involving vectors.