Final answer:
The magnitude of the sum of five vectors depends on the direction and magnitude of each vector, not on the number of vectors added. The Pythagorean theorem applies to the addition of vectors at right angles, and electric-field lines from a positive point charge do spread outward radially. Waves can superimpose regardless of whether they are in the same line or have different frequencies.
Step-by-step explanation:
The addition of five vectors \( a, b, c, d, \) and \( e \) does not necessarily result in a vector with a greater magnitude than if only two of the vectors were added. The resultant vector's magnitude depends on the direction and magnitude of each vector being added. When vectors are added, they could cancel each other out if they are in opposite directions, reducing the overall magnitude.
The Pythagorean theorem can be used 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 a true statement, as in such a case, the vectors form the legs of a right-angled triangle, and the resultant vector forms the hypotenuse.
The statement that electric-field lines from a positive point charge spread out radially and point outward is true. Electric-field lines are used to represent the direction of the electric field, which, for a positive charge, points away from the charge.
The amplitudes of waves add up when they are propagating in the same line, which implies constructive interference occurs. However, this statement does not imply that waves with different propagation lines cannot sum their amplitudes through the principle of wave superposition; hence, the original statement is false.
At any given point in space, only a single electric-field line can exist. This is because electric-field lines are not physical objects but representations of field direction. They cannot cross each other because at a crossing point, it would imply two different directions for the field, which is not possible.
A vector can indeed form the shape of a right angle triangle with its x and y components. This is the basis of vector resolution into perpendicular components and is a fundamental concept in vector analysis.
Finally, the statement that waves can superimpose if their frequencies are different is true. This is related to the principle of wave superposition where two or more waves can pass through the same point and combine to form a new wave pattern.