Questions: What is the unit vector of v = ⟨6, -9⟩?

Solution Steps

The vector is given as \( \mathbf{v} = \langle 6, -9 \rangle \).

The magnitude \( ||\mathbf{v}|| \) of the vector is calculated using the formula: \[ ||\mathbf{v}|| = \sqrt{6^2 + (-9)^2} = \sqrt{36 + 81} = \sqrt{117} = 10.816653826391969 \]

The unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is found by dividing each component of \( \mathbf{v} \) by its magnitude: \[ \mathbf{u} = \left\langle \frac{6}{||\mathbf{v}||}, \frac{-9}{||\mathbf{v}||} \right\rangle = \left\langle \frac{6}{10.816653826391969}, \frac{-9}{10.816653826391969} \right\rangle \] This results in: \[ \mathbf{u} \approx \langle 0.554700196225229, -0.8320502943378436 \rangle \]

Final Answer