Find \( \mathbf{u} \cdot \mathbf{v}, \mathbf{u} \cdot \mathbf{u},\|\mathbf{u}\|^{2},(\mathbf{u} \cdot \mathbf{v}) \mathbf{v} \), and \( \mathbf{u} \cdot(2 \mathbf{v}) \) \[ \mathbf{u}=\langle 7,11\rangle, \quad \mathbf{v}=\langle-2,3\rangle \]

Let's compute each of the requested operations step by step using the vectors: \[ \mathbf{u} = \langle 7, 11 \rangle, \quad \mathbf{v} = \langle -2, 3 \rangle \] ### 1. Dot Product \( \mathbf{u} \cdot \mathbf{v} \) The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \] Applying this to \( \mathbf{u} \) and \( \mathbf{v} \): \[ \mathbf{u} \cdot \mathbf{v} = (7)(-2) + (11)(3) = -14 + 33 = 19 \] ### 2. Dot Product \( \mathbf{u} \cdot \mathbf{u} \) This is the dot product of \( \mathbf{u} \) with itself: \[ \mathbf{u} \cdot \mathbf{u} = (7)(7) + (11)(11) = 49 + 121 = 170 \] ### 3. Squared Norm \( \|\mathbf{u}\|^{2} \) The squared norm of a vector is the same as its dot product with itself: \[ \|\mathbf{u}\|^{2} = \mathbf{u} \cdot \mathbf{u} = 170 \] ### 4. Scalar Multiplication \( (\mathbf{u} \cdot \mathbf{v}) \mathbf{v} \) First, compute \( \mathbf{u} \cdot \mathbf{v} = 19 \) (from step 1). Then multiply this scalar by vector \( \mathbf{v} \): \[ (\mathbf{u} \cdot \mathbf{v}) \mathbf{v} = 19 \cdot \langle -2, 3 \rangle = \langle 19 \times (-2), \, 19 \times 3 \rangle = \langle -38, 57 \rangle \] ### 5. Dot Product with Scaled Vector \( \mathbf{u} \cdot (2 \mathbf{v}) \) First, scale vector \( \mathbf{v} \) by 2: \[ 2 \mathbf{v} = 2 \cdot \langle -2, 3 \rangle = \langle -4, 6 \rangle \] Then, compute the dot product with \( \mathbf{u} \): \[ \mathbf{u} \cdot (2 \mathbf{v}) = (7)(-4) + (11)(6) = -28 + 66 = 38 \] **Alternatively**, using the property of dot products with scalars: \[ \mathbf{u} \cdot (2 \mathbf{v}) = 2 (\mathbf{u} \cdot \mathbf{v}) = 2 \times 19 = 38 \] ### Summary of Results 1. \( \mathbf{u} \cdot \mathbf{v} = 19 \) 2. \( \mathbf{u} \cdot \mathbf{u} = 170 \) 3. \( \|\mathbf{u}\|^{2} = 170 \) 4. \( (\mathbf{u} \cdot \mathbf{v}) \mathbf{v} = \langle -38, 57 \rangle \) 5. \( \mathbf{u} \cdot (2 \mathbf{v}) = 38 \)