Question

Vector v is multiplied by a scalar resulting in u. On a coordinate plane, vector v has origin (0, 0) and terminal point (-6, -7). Vector u has origin (0, 0) and terminal point (2, 2). What can be the interval that contains the scalar?

1) less than -1
2) between -1 and 0
3) between 0 and 1
4) greater than 1

Answer 1

The correct interval for the scalar is between -1 and 0 since it changes the signs of the components, indicating it is negative, and scales down the magnitude, indicating its absolute value is less than 1.

Step-by-step explanation:

When a vector v is multiplied by a scalar, resulting in a vector u, the new vector u will have its length scaled by the absolute value of that scalar, and if the scalar is negative, the direction of u will be opposite to that of v. Given that vector v has terminal point (-6, -7) and vector u has terminal point (2, 2), we can infer that the scalar multiplied by v to produce u changes the signs of the components of v, therefore, the scalar must be negative. We also know the magnitude of u is less than the magnitude of v, indicating that the absolute value of the scalar should be less than 1. Hence, for the scalar c, the interval that contains c must meet the condition that it is negative and has an absolute value less than 1, which means c belongs to the interval between -1 and 0.