Identify scalar/vector quantities and perform vector calculations given coordinates.

A scalar quantity has only magnitude. A vector quantity has both magnitude and direction.

• a) Time: Scalar
• b) Impulse: Vector
• c) Momentum: Vector
• d) Velocity: Vector
• e) Mass: Scalar
• f) Acceleration: Vector

A diagram would show points A(1,3), B(3,1), C(4,4), D(2,6) plotted on a coordinate plane. Vectors $\vec{AB}$, $\vec{BC}$, $\vec{CD}$, and $\vec{AD}$ would be drawn as arrows connecting these points.

The length (magnitude) of a vector $(p, q)$ is given by $\sqrt{p^2 + q^2}$.

• Length of $\vec{u}$: $$|\vec{u}| = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = \boxed{2\sqrt{2}}$$
• Length of $\vec{v}$: $$|\vec{v}| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \boxed{\sqrt{10}}$$
• Length of $\vec{w}$: $$|\vec{w}| = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = \boxed{2\sqrt{2}}$$
• Length of $\vec{x}$: $$|\vec{x}| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \boxed{\sqrt{10}}$$

Two vectors are parallel if one is a scalar multiple of the other.

• Compare $\vec{u} = (2, -2)$ and $\vec{w} = (-2, 2)$. Since $\vec{w} = -1 \cdot (2, -2) = -\vec{u}$, the lines $\vec{AB}$ and $\vec{CD}$ are parallel.
• Compare $\vec{v} = (1, 3)$ and $\vec{x} = (1, 3)$. Since $\vec{x} = 1 \cdot (1, 3) = \vec{v}$, the lines $\vec{BC}$ and $\vec{AD}$ are parallel.

The parallel pairs are $\boxed{\vec{AB} \text{ and } \vec{CD}}$, and $\boxed{\vec{BC} \text{ and } \vec{AD}}$.

Two vectors are equal if they have the same components (magnitude and direction).

• Comparing the component forms: $\vec{u} = (2, -2)$ $\vec{v} = (1, 3)$ $\vec{w} = (-2, 2)$ $\vec{x} = (1, 3)$ The vectors with identical components are $\vec{v}$ and $\vec{x}$. Therefore, $\boxed{\vec{v} = \vec{x}}$.