Final answer:
To find two vectors parallel to the given vector v with three times its magnitude, we multiply v by 3 to get (15, -21) and by -3 to get (-15, 21). The magnitude of the resultant vector of two equal and opposite vectors is zero.
Step-by-step explanation:
To find two vectors parallel to vector v with three times the magnitude of vector v, we need to multiply vector v by the scalar 3. Given vector v = (5, -7), the first parallel vector with three times the magnitude of vector v is obtained by calculating (3 × 5, 3 × -7), which gives us (15, -21). Similarly, another vector parallel to vector v with the same magnitude can be found, but since we require two distinct vectors, we can also consider the vector with the opposite direction by multiplying by -3, so the second vector is (-15, 21).
If two vectors are equal in magnitude and opposite in direction, the magnitude of their resultant vector is zero because they cancel each other out. When vectors are multiplied by a scalar, the resulting vector has a magnitude that is the absolute value of the scalar times the original magnitude, while the direction remains the same if the scalar is positive and is reversed if the scalar is negative. Vectors that are orthogonal to each other have a resultant magnitude that is the product of their magnitudes and the sine of 90°, which is 1, hence the resultant is also the product of their magnitudes.