Step 1: Determine the vectors

You are given the coordinates of the tail and head of two vectors:

• Arrow A: Tail = $(10, 0)$, Head = $(40, 40)$
• Arrow B: Tail = $(80, 0)$, Head = $(60, 20)$

To find the vector representation of each arrow, subtract the tail coordinates from the head coordinates:

1. Vector A: $\text{Vector A} = \left(40 - 10, 40 - 0\right) = (30, 40)$

2. Vector B: $\text{Vector B} = \left(60 - 80, 20 - 0\right) = (-20, 20)$

Step 2: Calculate the resultant vector

The resultant vector is found by adding the components of Vector A and Vector B:

$\text{Resultant vector} = \left(30 + (-20), 40 + 20\right) = (10, 60)$

Step 3: Calculate the magnitude of the resultant vector

The magnitude $|R|$ of a vector $(x, y)$ is given by the formula:

$|R| = \sqrt{x^2 + y^2}$

Substituting the components of the resultant vector $(10, 60)$:

$|R| = \sqrt{10^2 + 60^2} = \sqrt{100 + 3600} = \sqrt{3700} \approx 60.83$

So, the magnitude of the resultant vector is approximately 60.83 units.

Step 4: Calculate the angle of the resultant vector

The angle $\theta$ of the resultant vector with respect to the x-axis can be found using the formula:

$\theta = \tan^{-1} \left( \frac{y}{x} \right)$

Substitute the values for $x = 10$ and $y = 60$:

$\theta = \tan^{-1} \left( \frac{60}{10} \right) = \tan^{-1}(6)$

Using a calculator:

$\theta \approx 80.54^\circ$

So, the angle of the resultant vector is approximately 80.54 degrees.

Step 5: Graph representation

To visualize this, the vectors would be drawn on a coordinate plane:

1. Vector A starts from $(10, 0)$ and points to $(40, 40)$.
2. Vector B starts from $(80, 0)$ and points to $(60, 20)$.
3. The resultant vector, formed by adding these two, would start from the origin and point to $(10, 60)$.

Would you like me to generate a graph for you to visualize this?

Questions:

1. How do you calculate the magnitude of a vector?
2. What formula is used to calculate the angle of a vector?
3. How can you graphically represent vector addition?
4. Why is the resultant vector the sum of individual vectors' components?
5. What happens to the magnitude when you subtract two vectors?

Tip:

To calculate angles in different quadrants, always verify if the vector lies above or below the x-axis and adjust accordingly when using arctangent.