Final answer:
Two parallel vectors to v = 6, -8, 0 with a length of 15 are v1 = 9, -12, 0 and v2 = -9, 12, 0 which are both scaled versions of the unit vector in the direction of v.
Step-by-step explanation:
To find two vectors parallel to a given three-dimensional vector v with a specific length, you must first find a unit vector in the direction of v and then scale it to have the desired length. The vector is v = 6, -8, 0 and we want vectors of length 15.
The magnitude or length of vector v is calculated using the formula magnitude = √(x² + y² + z²), resulting in √(6² + (-8)² + 0²) = √(36 + 64) = √100 = 10.
To find the unit vector u in the direction of v, we divide each component of v by its magnitude.
u = (6/10, -8/10, 0/10) = (0.6, -0.8, 0)
Now scale the unit vector to get the desired length of 15:
v₁ = 15u = (15*0.6, 15*(-0.8), 15*0) = (9, -12, 0)
The second parallel vector of the same length but in the opposite direction is:
v₂ = -15u = (-15*0.6, -15*(-0.8), -15*0) = (-9, 12, 0)