Final answer:
A vector can indeed form a right angle triangle with its components, and the direction of the resultant vector depends on the magnitude and direction of the added vectors. The Pythagorean theorem is applicable for vectors at right angles to each other, and knowing the angles and magnitude of one vector, the resultant vector's magnitude and direction can be determined.
Step-by-step explanation:
A vector can indeed form the shape of a right-angle triangle with its x and y components. This is because any 2-D vector can be broken down into its horizontal (x) and vertical (y) components, which can be considered as the sides of a right-angle triangle with the vector itself being the hypotenuse.
It is also true that the direction of the resultant vector depends on both the magnitude and direction of the added vectors. The resultant is found by the vector addition of those vectors, which geometrically can be done using the head-to-tail method or algebraically using the components of the vectors.
Without knowledge of the magnitude of vectors, it is false that we can find the angle of the resultant vector if only the angles of two vectors are known, because the angle also depends on the magnitude of the vectors.
The Pythagorean theorem can indeed be used to calculate the length of the resultant vector when two vectors are at right angles to each other, making the statement true. The squares of the lengths of the vectors (considered as sides of a right-angle triangle) add up to the square of the length of the resultant vector (the hypotenuse).
Regarding the addition of five vectors a, b, c, d, and e, it is false that their addition always results in a vector with a greater magnitude than if only two of the vectors were added. The resultant's magnitude depends on both the direction and the magnitude of the individual vectors.
It is false that every 2D vector can be expressed as the product of its x and y components. The correct expressions for any given vector A should be Ax = A cos θ and Ay = A sin θ, where θ is the angle that the vector makes with the horizontal axis.
Lastly, it is true that if we know the angles of two vectors and the magnitude of one, we can find the magnitude and direction of the resultant vector. With trigonometry and vector addition principles, we can solve for the unknowns.