Question

For v=4i-8j , find unit vector u in the direction of v.

Answer 1

To find the unit vector \( \mathbf{u} \) in the direction of the vector \( \mathbf{v} = 4\mathbf{i} - 8\mathbf{j} \), we need to follow these steps:

1. **Find the magnitude of \( \mathbf{v} \)**:
\[
|\mathbf{v}| = \sqrt{(4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}
\]

2. **Divide the vector \( \mathbf{v} \) by its magnitude** to find the unit vector \( \mathbf{u} \):
\[
\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{4\mathbf{i} - 8\mathbf{j}}{4\sqrt{5}} = \frac{4}{4\sqrt{5}}\mathbf{i} - \frac{8}{4\sqrt{5}}\mathbf{j}
\]
Simplifying:
\[
\mathbf{u} = \frac{1}{\sqrt{5}}\mathbf{i} - \frac{2}{\sqrt{5}}\mathbf{j}
\]

So, the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is:
\[
\mathbf{u} = \frac{1}{\sqrt{5}}\mathbf{i} - \frac{2}{\sqrt{5}}\mathbf{j}
\]