To find the unit vector in the direction of vector b, we need to first calculate the magnitude of b and then divide b by its magnitude.

Step 1: Find the magnitude of b

The magnitude of a vector $\mathbf{b} = 3\hat{i} + 4\hat{j}$ is given by:

$|\mathbf{b}| = \sqrt{b_x^2 + b_y^2}$

where $b_x = 3$ and $b_y = 4$. So,

$|\mathbf{b}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Step 2: Find the unit vector in the direction of b

The unit vector in the direction of b is obtained by dividing b by its magnitude:

$\hat{b} = \frac{\mathbf{b}}{|\mathbf{b}|}$

Substitute the values:

$\hat{b} = \frac{3\hat{i} + 4\hat{j}}{5}$

$\hat{b} = \frac{3}{5} \hat{i} + \frac{4}{5} \hat{j}$

Thus, the unit vector in the direction of b is:

$\hat{b} = \frac{3}{5} \hat{i} + \frac{4}{5} \hat{j}$

Final Answer:

The unit vector in the direction of b is $\hat{b} = \frac{3}{5} \hat{i} + \frac{4}{5} \hat{j}$.

Would you like to go over more details or need help with another problem? Here are a few related questions:

1. How do you find the magnitude of a vector in 3D space?
2. How do you find the unit vector in the direction of vector a?
3. How do you calculate the angle between two vectors?
4. What is the cross product of two vectors and how do you calculate it?
5. How does the dot product of vectors relate to their magnitudes and the cosine of the angle between them?

Tip: Always remember that the unit vector is a vector with magnitude 1, which means it represents only the direction of the original vector, not its magnitude.