Scalar quantities are defined by a magnitude with no applicable direction. In contrast, vector quantities must have both magnitude and direction of action.

The product of a vector quantity and a scalar quantity will always be a vector quantity. Force results from the product of mass (scalar) and acceleration (vector). Weight is a type of force, generated by the acceleration of gravity.

Voltage is a scalar quantity and can be calculated by the product of current (scalar) and resistance (scalar).

Work is a vector quantity and can be calculated by the product of a force (vector) and displacement (vector).

Velocity is a vector and can be calculated by the quotient of displacement (vector) per unit time (scalar).

Scalar quantities are defined by a magnitude with no applicable direction. In contrast, vector quantities must have both magnitude and direction of action.

Speed is defined by a change in distance per unit time. Since distance and time are both scalar quantities, the resulting speed is also scalar. In contrast, velocity is given by a change in displacement per unit time. Since displacement is a vector, the resulting velocity is also a vector. The magnitude of a given speed and given velocity may be equal, but the velocity term will represent the speed applied in a certain direction.

Acceleration is a vector quantity determined by a change in velocity per unit time. Weight is generated by the force of gravity on an object; all forces are vectors.

Scalar quantities are defined by a magnitude with no applicable direction. In contrast, vector quantities must have both magnitude and direction of action.

Some common scalar quantities are distance, speed, mass, and time. Some common vector quantities are force, velocity, displacement, and acceleration.

A vector has both magnitude AND direction, while a scalar just has a magnitude. When asking if something is a vector or a scalar, ask if a direction would make sense -- in this case, force is the only vector. While a direction would help with speed and distance, those are both scalars; the vector version of speed is velocity, and the vector version of distance is displacement.

A scalar quantity can be defined by magnitude alone, while a vector quantity must be defined by both magnitude and direction of action.

Of the given answer options, mass if the only scalar quantity. Mass has magnitude, generally in kilograms, but cannot act in a direction. "7kg west," for example, is nonsensical.

In contrast, displacement, velocity, force, and momentum must be applied in a given direction. Displacement is the vector equivalent of the scalar quantity distance, and velocity is the vector equivalent of the scalar quantity speed. Forces must always act in a given direction, and have no scalar equivalent. Similarly, momentum must always be directional.

There is a distinct and crucial difference between measuring displacement and measuring distance. Distance is a scalar quantity, which means that it depends on the path taken and is independent of the direction traveled. Distance measures the total length traveled, without any reference to the starting point.

In contrast, displacement is a vector quantity. This means that both the magnitude of the length and its direction must be factored into the calculation. Displacement is essentially the net distance traveled in relation to the starting point, independent of the path traveled.

In this question, the skater finishes in exactly the same place that she started. Without any other information, we can conclude that her displacement is zero. It does not matter what path she took to return to her starting point; she could have taken one step forward and one step back, skated the entire rink seventeen times, or simply jumped and landed. All of these possibilities would result in zero displacement.

Viscosity is the measurement of a "thickness" of a liquid. Molasses, for example, is a more viscous fluid than water is.

Vector measurements are defined by a magnitude and a direction. For a liquid to have a measureable "thickness" is logical, but a liquid cannot have a viscosity in a direction. To say that a fluid has a viscosity of East makes no sense. Viscosity is a scalar quantity.

Displacement, force, velocity, and acceleration all have associated directions and are classified as vector quantities.

Vector quantities are defined by both the magnitude of the parameter and the direction of action. In contract, scalar quantities are independent of direction and rely only on the magnitude of the parameter.

Mass, distance, time, and brightness are all scalar quantities. This is to say that none of these terms can be applied in a given direction. It would be illogical to have "three grams west" or "eighteen seconds to the left." Distance is the scalar equivalent of the displacement vector.

Force is always a vector quantity, since the direction of the force matters in defining the parameter. "Four Newtons to the right" is quantifiably different from "four Newtons downward" or "four Newtons to the left."

The magnitude of the vector is found using the distance formula:

To calculate the angle we must first find the inverse tangent of :

This angle value is the principal arctan, but it is in the fourth quadrant while our vector is in the second. We must add the angle 180° to this value to arrive at our final answer.

For vector , the magnitude is doubled, but the direction remains the same.

For our calculation, we use a magnitude of:

The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.

The resultant vector is: .