Final answer:
It is false that knowing only the angles of two vectors allows for the calculation of the resultant vector angle; the magnitude is also needed. Short-stitch cotton was not preferred before the cotton gin, which made its processing more viable. The Pythagorean theorem applies to vectors at right angles, and a vector can form a right-angle triangle with its components.
Step-by-step explanation:
When analyzing vectors in high school physics, there are several key concepts to understand to accurately determine the properties of the resultant vectors. First, addressing the GRASP CHECK question, whether it's true or false that if only the angles of two vectors are known, we can find the angle of their resultant addition vector, the answer is false. Knowing only the angles is insufficient; one must also know the magnitude of the vectors to find the resultant.
Addressing the notion of cotton staple length in question Exercise 11.3.3, it is false that short-staple cotton was preferred to long-staple cotton before the invention of the cotton gin. The cotton gin made short-stitch cotton more economically viable by easing the removal of seeds.
Furthermore, the Pythagorean theorem can indeed be used to calculate the length of the resultant vector obtained from the addition of two vectors that are at right angles to each other, as stated in question 36. Lastly, it is also true that a vector can form the shape of a right-angle triangle with its x and y components, as indicated in question 60, facilitating the use of trigonometric methods to analyze the vector.