Final answer:
It is false that any 5 vectors that span a 5-dimensional space V automatically form a basis for V; they also need to be linearly independent. Several true/false statements related to vectors are explained with an emphasis on the principles of magnitude, linear independence, and vector addition.
Step-by-step explanation:
The statement that "any 5 vectors that span a 5-dimensional vector space V form a basis for V" is false. While it is true that a basis for a vector space V must span V and must consist of the same number of vectors as the dimension of V, it is also required that the vectors in a basis must be linearly independent. If the vectors are not linearly independent, they do not form a basis even if they span the space.
For the specific questions related to vectors:
- Adding five vectors a, b, c, d, and e does not necessarily result in a vector with a magnitude greater than the magnitude of any two vectors added together. The resultant vector's magnitude depends on both the magnitudes and directions of the individual vectors.
- A vector can indeed form the shape of a right angle triangle with its x and y components, which is a true statement.
- It is false that every 2-D vector can be expressed solely as the product of its x and y components, because a vector also has a direction, not just magnitude components.
- If only the angles of two vectors are known, without their magnitudes, we cannot find the exact angle of their resultant vector.
- With the knowledge of the angles of two vectors and the magnitude of one, we can find the magnitude and direction of the resultant vector using vector addition principles.
- The direction of the resultant vector indeed depends on both the magnitude and direction of the added vectors, which makes this statement true.