Compute the dot product of the vectors \( \mathbf{v} \) and \( \mathbf{w} \), and find the angle between the vectors. \( \mathbf{v}=-9 \mathbf{i}-\mathbf{j} \) and \( \mathbf{w}=-\mathbf{i}-9 \mathbf{j} \) \( \mathbf{v} \cdot \mathbf{w}=18 \) (Simplify your answer.) Find the magnitude of the vectors. \( |\mathbf{v}|=\square \) and \( |\mathbf{w}|=\square \) (Type exact answers, using radicals as needed.)

To find the magnitudes of the vectors \( \mathbf{v} \) and \( \mathbf{w} \), we'll use the formula for the magnitude of a vector in two dimensions: \[ |\mathbf{a}| = \sqrt{a_x^2 + a_y^2} \] where \( a_x \) and \( a_y \) are the components of vector \( \mathbf{a} \). ### Calculating \( |\mathbf{v}| \) Given \( \mathbf{v} = -9\mathbf{i} - \mathbf{j} \), the components are: - \( v_x = -9 \) - \( v_y = -1 \) \[ |\mathbf{v}| = \sqrt{(-9)^2 + (-1)^2} = \sqrt{81 + 1} = \sqrt{82} \] ### Calculating \( |\mathbf{w}| \) Given \( \mathbf{w} = -\mathbf{i} - 9\mathbf{j} \), the components are: - \( w_x = -1 \) - \( w_y = -9 \) \[ |\mathbf{w}| = \sqrt{(-1)^2 + (-9)^2} = \sqrt{1 + 81} = \sqrt{82} \] ### Summary \[ |\mathbf{v}| = \sqrt{82} \quad \text{and} \quad |\mathbf{w}| = \sqrt{82} \]