Vector u has a magnitude of 26 and vector v has a magnitude of 58. The angle between the two vectors is 77 degrees.
a) Draw a fully-labelled vector diagram showing the two vectors and the resultant vector when they are added together.
b) Find the magnitude of the resultant vector.
c) Find the direction of the resultant vector relative to vector u.

Let's break down the problem step by step.

### a) Drawing the fully-labeled vector diagram
1. **Vectors u and v**: Draw vector **u** with a magnitude of 26, and vector **v** with a magnitude of 58. The angle between these two vectors is 77°.
2. **Resultant vector (u + v)**: The resultant vector represents the sum of vectors **u** and **v**. To draw the resultant:
   - Place **v**'s tail at **u**'s head.
   - The resultant vector starts at **u**'s tail and ends at **v**'s head.
3. **Labeling**: Label both vector **u**, vector **v**, and the angle of 77° between them. Also, label the resultant vector.

I can provide a drawing for this setup, or if you need a detailed textual description, let me know!

### b) Finding the magnitude of the resultant vector
We can use the **Law of Cosines** to find the magnitude of the resultant vector, denoted as \( R \).

The formula is:
\[
R = \sqrt{u^2 + v^2 + 2uv \cos(\theta)}
\]
where:
- \( u = 26 \) (magnitude of vector u)
- \( v = 58 \) (magnitude of vector v)
- \( \theta = 77^\circ \) (angle between vectors u and v)

Substituting the values:

\[
R = \sqrt{26^2 + 58^2 + 2(26)(58) \cos(77^\circ)}
\]

Let's calculate this.

The magnitude of the resultant vector is approximately \( 68.69 \) units.

### c) Finding the direction of the resultant vector relative to vector u
To find the direction (angle \( \alpha \)) of the resultant vector relative to vector **u**, we can use the **Law of Sines** or the formula derived from the components of the vectors. Using vector components, we can find:

\[
\alpha = \arctan\left(\frac{v \sin(\theta)}{u + v \cos(\theta)}\right)
\]

Let's compute this angle.The direction of the resultant vector relative to vector **u** is approximately \( 55.36^\circ \).

### Summary:
- **a)** The vector diagram consists of vectors **u** and **v** with magnitudes of 26 and 58, respectively, making an angle of 77°, and the resultant vector being the sum of both.
- **b)** The magnitude of the resultant vector is approximately \( 68.69 \) units.
- **c)** The direction of the resultant vector relative to vector **u** is approximately \( 55.36^\circ \).

Would you like a detailed drawing for part (a), or do you have any further questions?

### Related Questions:
1. How would the resultant magnitude change if the angle between the vectors was smaller?
2. Can you break down the resultant vector into horizontal and vertical components?
3. How does changing the magnitude of vector **u** affect the resultant vector?
4. What is the maximum possible resultant magnitude given two vectors?
5. How can the Law of Cosines be applied to real-world vector problems?

### Tip: 
When adding vectors graphically, the **tip-to-tail** method provides a clear way to visualize the resultant vector.

Vector u has a magnitude of 26 and vector v has a magnitude of 58. The angle between the two vectors is 77 degrees.
 a) Draw a fully-labelled vector diagram showing the two vectors and the resultant vector when they are added together.
 b) Find the magnitude of the resultant vector.
 c) Find the direction of the resultant vector relative to vector u.

Learn how to calculate the magnitude and direction of a resultant vector using the Law of Cosines. This problem involves vectors with magnitudes 26 and 58, and an angle of 77 degrees. Detailed step-by-step solution provided, suitable for Grades 10-12.

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