Let’s solve the problem step by step:

Given:

The two vectors $\vec{u}$ and $\vec{v}$ need to be added to find the resultant vector $\vec{A} = \vec{u} + \vec{v}$.

From the diagram:

1. Vector $\vec{u}$:

   • Starts at the origin and ends at $(3, 2)$.
   • Therefore, $\vec{u} = (3, 2)$.

2. Vector $\vec{v}$:

   • Starts at the origin and ends at $(-1, -4)$.
   • Therefore, $\vec{v} = (-1, -4)$.

(a) Drawing the resultant vector:

The resultant vector $\vec{A}$ is found by adding $\vec{u}$ and $\vec{v}$: $\vec{A} = \vec{u} + \vec{v} = (3, 2) + (-1, -4) = (3 - 1, 2 - 4) = (2, -2).$

To draw it:

• Start at the origin $(0, 0)$.
• Draw the arrow to the point $(2, -2)$.

(b) Calculating the length and direction of $\vec{A}$:

1. Length of $\vec{A}$: The length (magnitude) of $\vec{A}$ is: $|\vec{A}| = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}.$

2. Direction of $\vec{A}$: The direction (angle $\theta$ with the positive x-axis) is given by: $\theta = \tan^{-1}\left(\frac{\text{y-component}}{\text{x-component}}\right) = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1).$ Since $\tan^{-1}(-1) = -45^\circ$, the angle is $-45^\circ$ (measured clockwise from the positive x-axis).

Final Answer:

1. Resultant Vector: $\vec{A} = (2, -2)$.
2. Length of $\vec{A}$: $2\sqrt{2} \approx 2.83$ units.
3. Direction of $\vec{A}$: $-45^\circ$ or $315^\circ$ (if expressed as a positive angle).

Let me know if you'd like a detailed explanation of any part or the drawing.

Related Questions:

1. How do you find the magnitude of a vector given its components?
2. What is the physical interpretation of vector addition?
3. How do you determine the quadrant of a vector's angle?
4. What is the geometric method of adding two vectors (parallelogram rule)?
5. How do you convert between degrees and radians for angles?

Tip:

Always draw vectors head-to-tail to visualize addition and subtraction effectively.