To find the unit vector that has the same direction as the given vector $\mathbf{v} = -8\mathbf{i} + 6\mathbf{j}$, we use the following steps:

Step 1: Compute the magnitude of the vector $\mathbf{v}$

The formula for the magnitude of a vector $\mathbf{v} = a\mathbf{i} + b\mathbf{j}$ is:

$\|\mathbf{v}\| = \sqrt{a^2 + b^2}$

Substitute $a = -8$ and $b = 6$:

$\|\mathbf{v}\| = \sqrt{(-8)^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$

Step 2: Divide the vector by its magnitude

The unit vector $\mathbf{u}$ is given by:

$\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$

Substitute $\mathbf{v} = -8\mathbf{i} + 6\mathbf{j}$ and $\|\mathbf{v}\| = 10$:

$\mathbf{u} = \frac{-8\mathbf{i} + 6\mathbf{j}}{10} = -0.8\mathbf{i} + 0.6\mathbf{j}$

Final Answer:

The unit vector is:

$\mathbf{u} = -0.8\mathbf{i} + 0.6\mathbf{j}$

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Would you like additional details or have further questions?

Related Questions:

1. How do you verify that a vector is a unit vector?
2. What are the applications of unit vectors in physics?
3. How do you calculate the direction of a vector using trigonometry?
4. What happens to a unit vector's components if the original vector is scaled?
5. Can unit vectors exist in three-dimensional space, and how are they calculated?

Tip:

Always ensure you normalize a vector correctly by dividing by its magnitude, as this ensures the resultant vector has a magnitude of 1