Final answer:
The expression cos(a) + cos^2(a) is not always positive; it depends on the value of a. The GRASP CHECK statements are notably false regarding vectors' angles only, true for resultant vectors at right angles, and true for vectors forming right angle triangles with components.
Step-by-step explanation:
The statement that cos(a) + cos2(a) is always positive is false. For example, consider an angle of a = 180 degrees (or π radians). In this case, cos(180°) = -1, and cos2(180°) = 1, which adds up to 0, neither positive nor negative. However, for angles between 0 and 90 degrees, where the cosine function is positive, the expression indeed remains positive. But we cannot generalize for all angles.
For the GRASP CHECK:
- False - Knowing only the angles of two vectors is not enough to find the angle of their resultant addition vector without additional information.
- True - The Pythagorean theorem can be used to calculate the length of the resultant vector when two vectors are at right angles to each other.
- True - A vector can indeed form the shape of a right angle triangle with its x and y components.